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Al^  ELEMENTARY  COURSE 


IN 


ANALYTIC    GEOMETRY 


BY 

Q 

J.    H.    TANNER 

ASSISTANT  PROFESSOR  OF  MATHEMATICS   IN  CORNELL  UNIVERSITY 

AND 

JOSEPH   ALLEN 

FORMERLY   INSTRUCTOR   IN   MATHEMATICS   IN   CORNELL    UNIVERSITY 
TUTOR   IN   THE   COLLEGE   OF  THE   CITY   OF  NEW   YORK 


BOSTON  COLLEGE  LIFKAHT 
^:.o_    CHESTOTT  HILL,  M^.^. 


MATH,  DEPT, 


NEW  YORK  •:•  CINCINNATI  •;•  CHICAGO 

AMERICAN    BOOK    COMPANY 


Copyright,  1898,  by 
J.  H.  TANNER  axd  JOSEPH   ALLEN. 


ANA.  GEOM. 
W.    P.    I 


15055ii 


THE   CORNELL    MATHEMATICAL    SERIES 

LUCIEN  AUGUSTUS    WAIT  •  •  •  General  Editor 

(SENIOE  PROFESSOR  OF  MATHEMATICS  IN  CORNELL  UNIVERSITY) 


The  Cornell  Mathematical  Series, 
lucien  augustus  wait, 

(Senior  Professor  of  Mathematics  in  Cornell  University,) 
GENERAL  EDITOR. 


This  series  is  designed  primarily  to  meet  tlie  needs  of  students  in  En- 
gineering and  Architecture  in  Cornell  University  ;  and  accordingly  many 
practical  problems  in  illustration  of  the  fundamental  principles  play  an  early 
and  important  part  in  each  book. 

While  it  has  been  the  aim  to  present  each  subject  in  a  simple  manner, 
yet  rigor  of  treatment  has  been  regarded  as  more  important  than  simplicity, 
and  thus  it  is  hoped  that  the  series  will  be  acceptable  also  to  general  students 
of  Mathematics. 

The  general  plan  and  many  of  the  details  of  each  book  were  discussed  at 
meetings  of  the  mathematical  staff.  A  mimeographed  edition  of  each  vol- 
ume was  used  for  a  term  as  the  text-book  in  all  classes,  and  the  suggestions 
thus  brought  out  were  fully  considered  before  the  work  was  sent  to  press. 

The  series  includes  the  following  works  : 
ANALYTIC  GEOMETRY.     By  J.  H.  Tanner  and  Joseph  Allen. 

DIFFERENTIAL  CALCULUS.     By  James 'McMahon  and  Virgil  Snyder. 

INTEGRAL  CALCULUS.    By  D.  A.  Murray. 


PREFACE 

Although  in  the  writing  of  this  book  the  needs  of  the 
students  in  the  various  departments  of  Engineering  and  of 
Architecture  in  Cornell  University  have  received  the  first 
consideration,  care  has  also  been  taken  to  make  the  work 
suitable  for  the  general  student  and  at  the  same  time  useful 
as  an  introduction  to  a  more  advanced  course  for  those 
students  who  may  wish  to  specialize  later  in  mathematics. 

Among  the  features  of  the  book  are : 

(1)  An  extended  introduction  (Chaps.  II,  III,  IV),  in 
which  it  is  hoped  that  the  fundamental  problems  of  the  sub- 
ject are  clearly  set  forth  and  sufficiently  illustrated.  The 
chief  difficulty  which  the  beginner  in  Analytic  Geometry 
usually  has  to  overcome  is  the  relation  between  an  equation 
and  its  locus ;  having  really  mastered  this,  he  easily  and 
rapidly  acquires  a  knowledge  of  the  properties  to  which  this 
relation  leads,  and  especial  care  has  therefore  been  given  to 
this  matter.  Analytic  Geometry  is  broader  than  Conic  Sec- 
tions, and  it  is  the  firm  conviction  of  the  authors  that  it  is 
far  more  important  to  the  student  that  he  should  acquire  a 
familiarity  with  the  spirit  of  the  method  of  the  subject  than 
that  he  should  be  required  to  memorize  the  various  properties 
of  any  particular  curve. 

(2)  The  making  use  of  some  intrinsic  properties  of  curves 
(see  Arts.  106,  112,  118),  which  experience  with  many 
classes  has  shown  to  give  the  student  an  unusually  strong 
grasp  on  the  equation  of  the  second  degree  from  Avhich  the 
a:z/-term  is  absent. 

(3)  Introduction  of  the  demonstrations  of  general  theorems 
by  numerical  examples.  This  not  only  makes  clear  to  the 
student  what  is  to  be  done,  but  shows  also  the  method  to 
be  employed,  —  it  generalizes  after  the  student  is  acquainted 
with  the  particular. 

(4)  Easy  but  rigorous  proofs  of  all  the  theorems  within 
the  scope  of  the  book.     E.g.,  in  Art.  67  it  is  proved,  and 


vi  PBEFACE 

very  simply,  too,  that  tlie  vanishing  of  the  discriminant  is 
not  only  a  necessary^  but  also  the  sufficient  condition  that  the 
quadratic  equation  represents  a  pair  of  straight  lines. 

It  may  also  be  mentioned  here  that,  in  the  early  part  of 
the  book,  two  or  more  figures  are  given  in  connection  with 
a  proof  and  so  lettered  that  the  same  demonstration  applies 
to  each.  It  is  hoped  that  this  will  help  to  convince  the 
student  of  the  generality  of  the  demonstration.  A  copious 
index  which  enables  one  almost  instantly  to  turn  to  anything 
contained  in  the  book  has  also  been  added. 

The  engineering  students  at  Cornell  University  study 
Analytic  Geometry  during  the  first  term  of  their  freshman 
year,  and  experience  has  shown  that  it  is  best  to  devote  a 
few  lessons  at  the  beginning  of  the  term  to  a  rapid  review 
of  those  parts  of  the  Algebra  and  Trigonometry  that  are 
essential  to  the  reading  of  the  Analytic  Geometry.  The  first 
twenty-three  pages  are  devoted  to  this  matter,  and  may,  of 
course,  be  omitted  by  those  classes  that  take  up  the  subject 
immediately  after  reading  the  Algebra  and  Trigonometry. 

The  book  contains  little  more  than  can  be  mastered  by  a 
properly  prepared  student  of  average  ability  in  from  twelve 
to  fourteen  weeks  ;  if  less  than  that  time  can  be  devoted  to 
the  work,  the  individual  teacher  will  know  best  what  parts 
may  be  most  wisely  omitted  by  his  pupils.  A  list  of  lessons 
for  a  short  course  of  eleven  weeks  is,  however,  suggested  on 
the  next  two  pages. 

A  few  specific  acknowledgments  of  indebtedness  are  made 
in  foot-notes  in  the  appropriate  places  in  the  book.  Of  the 
large  number  of  examples  which  are  inserted,  many  are  origi- 
nal, while  many  others  have  come  to  be  so  common  in  text- 
books that  no  specific  acknowledgment  for  them  can  be 
made.  We  take  great  pleasure  in  expressing  here  our 
thanks  to  the  other  authors  of  this  series  of  books  for  their 
many  helpful  suggestions  and  criticisms ;  to  our  colleagues. 
Dr.  J.  I.  Hutchinson  and  Dr.  G.  A.  Miller,  who  have  so 
greatly  assisted  us  in  reading  the  proof,  and  the  latter  of 
whom  also  read  the  manuscript  before  it  went  to  press ; 
to  Mr.  Peter  Field,  Fellow  in  Mathematics,  and  Mr.  E. 
A.  Miller  for  solving  the  entire  list  of  examples  ;  and  to 
Mr.  V.  T.  Wilson,  Instructor  in  Drawing  in  Sibley  College, 
for  the  care  with  which  he  has  made  the  figures. 


LIST   OF   LESSONS    SUGGESTED   FOR   A 
SHORT   COURSE 

[From  the  various  sets  of  exercises  the  teacher  is  expected  to  make  selec- 
tions lor  each  lesson.  The  fifth  day  of  each  week  should  be  devoted  to 
reviewing  the  preceding  four  lessons.] 

Lesson  Pages  Articles 

1  .      o      .      .      .  1-9        .....  1-8 

2  .     .     .    •.     .  9-15     .....  9-12 

3  ....     .    15-23 13-17 

4 24-28 18-22 

5 29-33 23-27 

6 34-40 28-30 

7 40-42     .....  31 

8  ....     .    43-52 32-37 

As  far  as  "  Exercises,"  p.  52. 

9     52-57     .....  38-41 

10  ....     .  58-60 

11 61-65 42-45 

12 65-73 46-48 

With  examples  selected  from  p.  79. 

13  ....     .  73-80     .....  49 

14 81-85 50-53 

15 86-94 54-58 

16 94-98     .....  59-61 

17 98-104 62-63 

18  ....     .  105-110  .....  64-65 

19  ....    I  11^-11^  .     ...  66,  67,  69 

1  118-119  '      ' 

20  ...     .     .119-122 

2^  r 123-127  r  70-72 

1 129-131  •     •     •     •       1 75-76 
vii 


viii  LESSONS  FOB   SHORT  COURSE 

Lesson  Pages                                 Articles 

22 131-137 77-78 

23 137-142 79-82 

24 142-149 83-85 

25 149-155  .....  86-90 

2Q 156-165 93-100 

27 165-169 

28  ....  .  170-177 101-107 

29 179-186 109-112 

^«  •  •  •  •  [f^Z-  ■  ■ '''''''-''' 

31 195-202 118-122 

32 203-208 123-126 

33 209-216 127-132 

34 216-218 

35 219-225 133-137 

36 225-233 138-140 

37 235-242 142-145 

33  f  242-247         (  146-148 
'  '  *  '  1 250-254  •  •  •  •  j 152-154 

39  ....  .  254-264 155-157 

40 265-272 160-164 

41 272-283 165-170 

42 284-291 171-174 

43 292-298 175-177 

44 309-330 185-198 


CONTENTS 


PART   I.— PLANE   ANALYTIC    GEOMETRY 


CHAPTER  I 

Introduction 
Algebraic  and  Trigonometric  Conceptions 

AKTICLB  PAGE 

1.  Number 1 

2.  Constants  and  variables 2 

3.  Functions 3 

4.  Identity,  equation,  and  root 4 

5.  Functions  classified 4 

6.  Notation 5 

7.  Continuous  and  discontinuous  functions 6 

8      ) 

'     y  The  quadratic  equation.     Its  solution 9 

10.  Zero  and  infinite  roots           ........  11 

11.  Properties  of  the  quadratic  equation 12 

12.  The  quadratic  equation  involving  two  unknowns       .         .         .13 

Trigonometric  Conceptions  and  Formulas 

13.  Directed  lines.     Angles 15 

14.  Trigonometric  ratios    .........  17 

15.  Functions  of  related  angles 18 

16.  Other  important  formulas 19 

17.  Orthogonal  projection .        .21 

CHAPTER  II 

Geometric  Conceptions.    The  Point 

I.    Coordinate  Systems 

18.  Coordinates  of  a  point 24 

19.  Analytic  Geometry 25 

ix 


CONTENTS 


AETICLE 

20.  Positive  and  negative  coordinates 

21.  Cartesian  coordinates  of  points  in  a  plane  . 

22.  Rectangular  coordinates       .         .         .        . 

23.  Polar  coordinates  ,         .         .        .        . 

24.  Notation 


25 
26 


:[ 


27. 
28. 
29. 


30. 
31. 


II.    Elementary  Applications 
Distance  between  two  points 

(1)  Polar  coordinates 

(2)  Cartesian  coordinates ;  axes  not  rectangular  . 

(3)  Rectangular  coordinates  .... 

Slope  of  a  line 

Summary       . 

The  area  of  a  triangle 

(1)  Rectangular  coordinates  .... 

(2)  Polar  coordinates 

To  find  the  coordinates  of  the  point  which  divides,  in  a 
ratio,  the  straight  line  from  one  given  point  to  another 
Fundamental  problems  of  analytic  geometry 


given 


PAGE 

25 

26 
27 
29 
30 


31 

32 
33 
33 
34 

34 
36 

37 
40 


CHAPTER  III 
The  Locus  of  an  Equation 

32.  The  locus  of  an  equation      .... 

33.  Illustrative  examples  :  Cartesian  coordinates 

34.  Loci  by  polar  coordinates     .... 

35.  The  locus  of  an  equation     .... 

36.  Classification  of  loci 

37.  Construction  of  loci.     Discussion  of  equations 

38.  The  locus  of  an  equation  remains  unchanged  :  (a)  by  any  trans- 

position of  the  terms  of  the  equation ;  and  (^)  by  multiply- 
ing both  members  of  the  equation  by  any  finite  constant 

39.  Points  of  intersection  of  two  loci 

40.  Product  of  two  or  more  equations 

41.  Locus  represented  by  the  sum  of  two  equations 

CHAPTER  IV 

The  Equation  of  a  Locus 

42.  The  equation  of  a  locus        ........ 

43.  Equation  of  straight  line  through  two  given  points    . 


43 
43 

46 
47 

48 
49 


52 
53 
54 
56 


61 
61 


CONTENTS  XI 

ARTICLE  PAGE 

44.     Equation   of  straight  line  through   given  point  and  in  given 

63 
64 
65 
66 
67 
73 


direction 

45.  Equation  of  a  circle ;  polar  coordinates 

46.  Equation  of  locus  traced  by  a  moving  point 

47.  Equation  of  a  circle  :  second  method 

48.  The  conic  sections         ..... 

49.  The  use  of  curves  in  applied  mathematics  . 


CHAPTER  V 
The  Straight  Line.     Equation  of  First  Degree 

Ax  +  By+C  =  0 

50.  Recapitulation 81 

51.  Equation  of  straight  line  through  two  given  points    .         .         .81 

52.  Equation  of  straight  line  in  terms  of  the  intercepts  which  it 

makes  on  the  coordinate  axes  .......     83 

53.  Equation  of  straight  line  through  a  given  point  and  in  a  given 

direction 84 

54.  Equation  of  straight  line  in  terms  of  the  perpendicular  from 

the  origin  upon  it,  and  the  angle  which  that  perpendicular 
makes  with  the  a:-axis        .         .         .         .         .         .         .         .86 

55.  I^ormal  form  of  equation  of  straight  line  :  second  method  .     87 

56.  Summary       ...........     88 

57.  Every  equation  of  the  first  degree  between  two  variables  has 

for  its  locus  a  straight  line 89 

58.  Reduction  of  the  general  equation  Ax  -{  By  -\-  C  =  0  to  the 

standard  forms.     Determination  of  a,  b,  m,  jj,  and  a  in  terms 

of  .4,  A  and  C 91 

59.  To  trace  the  locus  of  an  equation  of  the  first  degree  .         .         .94 

60.  Special  cases  of  the  equation  of  the  straight  line  Ax-]-By+  C  =  0     95 

61.  To  find  the  angle,  made  by  one  straight  line  with  another  .         .     97 

62.  Condition  that  two  lines  are  parallel  or  perpendicular         .         .     98 

63.  Line  which  makes  a  given  angle  with  a  given  line      .         .         .101 

64.  The  distance  of  a  given  point  from  a  given  line  .         .         .  105 

65.  Bisectors  of  the  angles  between  two  given  lines  .         .         .  108 

66.  The  equation  of  two  lines 110 

67.  Condition  that  the  general  quadratic  expression  may  be  factored  111 

68.  Equations  of  straight  lines  :  coordinate  axes  oblique  .         .         .115 

69.  Equations  of  straight  lines  :  polar  coordinates    ....  118 


xii  CONTENTS 

CHAPTER  VI 

Transformation  of  Coordinates 

ARTICLE  PAGE 

70.  Introductory 123 

I.    Cartesian  Coordinates  Only 

71.  Change  of  origin,  new  axes  parallel  respectively  to  the  original 

axes 124 

72.  Transformation    from    one    system    of    rectangular    axes    to 

another  system,  also  rectangular,  and  having  the  same  ori- 
gin ;  change  of  direction  of  axes  126 

73.  Transformation  from  rectangular  to  oblique  axes,  origin  un- 

changed   ...........     127 

74.  Transformation  from  one  set  of  oblique  axes  to  another,  origin 

unchanged        .         .         .         .         .         .         .         .         .         .     128 

75.  The  degree  of  an  equation  in  Cartesian  coordinates  is  not 

changed  by  transformation  to  other  axes        .         .         .         .     129 

^  II.    Polar  Coordinates 

76.  Transformations  between  polar  and  rectangular  systems  .         .     130 

CHAPTER  VII 

The  Circle 
Special  Equation  of  the  Second  Degree 

Ax"^  +  Aif  +  2  Gx  -V  2  Fy  ^  C  =  0 

77.  Introductory 135 

78.  The  circle  :  its  definition  and  equation 135 

79.  In  rectangular  coordinates  every  equation  of  the  form  x^  4-  y^ 

+  2Gx-\-2Fy-]-C  =  0  represents  a  circle     ....  137 

80.  Equation  of  a  circle  through  three  given  points        .         .         .  138 

Secants,  Tangents,  and  Normals 

81.  Definitions  of  secants,  tangents,  and  normals    .         .         .         .     140 

82.  Tangents :  IllustratiA^e  examples         ......     141 

83.  Equation  of  tangent  to  the  circle  x^  +  y'^  =  r^  in  terms  of  its 

slope 142 

84.  Equation  of  tangent  to  the  circle  in  terms  of  the  coordinates 

of  the  point  of  contact :  the  secant  method  ....     144 


CONTENTS 


Xlll 


85. 
86. 

87. 


89. 

90. 
91. 
92. 
93. 
94. 

95. 
96. 
97. 
98. 
99. 
100. 


PAGE 

Equation  of  a  normal  to  a  given  circle     .        .        ,        .         .  147 
Lengths  of   tangents   and  normals.     Subtangents    and   sub- 
normals ...........  149 

Tangent  and  normal  lengths,  subtangent  and  subnormal,  for 

the  circle 150 

To  find  the  length  of  a  tangent  from  a  given  external  point 
to  a  given  circle      .         .         .         .         .         .         .         •         .151 

From  any  point  outside  of  a  circle  two  tangents  to  the  circle 

can  be  drawn          . 152 

Chord  of  contact        .........  154 

Poles  and  polars 156 

Equation  of  the  polar        .         .         .         .         .         .         .         .  156 

Fundamental  theorem 157 

Geometrical  construction  for  the  polar  of  a  given  point,  and 

for  the  pole  of  a  given  line,  with  regard  to  a  given  circle   .  158 

Circles  through  the  intersections  of  two  given  circles      .         .  160 

Common  chord  of  two  circles   .......  160 

Radical  axis ;  radical  center      .......  161 

The  equation  of  a  circle  :  polar  coordinates     ....  162 

Equation  of  a  circle  referred  to  oblique  axes  ....  163 

The  angle  formed  by  two  intersecting  curves  ....  164 


CHAPTER  VIII 

The  Conic  Sections 

101.  Recapitulation 170 

I.    The  Parabola 
Special  Equation  of  Second  Degree 

Ax^  +  2Gx  +  2Fi/  +  C  =  0,  ov  Bf  +  2Gx-h2Fu  -^  a  =  0 

102.  The  parabola  defined 

103.  First  standard  form  of  the  equation  of  the  parabola 

104.  To  trace  the  parabola  y'^  =  4:px         .... 

105.  Latus  rectum     ........ 

106.  Geometric  property  of  the  parabola.     Second  standard  equa- 

tion        ........... 

107.  Every  equation  of  the  form 

Ax^-\-2Gx  +  2Fi/  +  C  =  0,  or  By^  +  2  Gx  ^  2  Fy  +  C  =  0, 

represents  a  parabola  whose  axis  is  parallel  to  one  of  the 
coordinate  axes       ......... 

108.  Reduction  of  the  equation  of  a  parabola  to  a  standard  form  . 


170 
171 
172 
173 

173 


175 

177 


xiv  CONTENTS 

II.    The  Ellipse 

Special  Equation  of  the  Second  Degree 

Ax^  +  Bif-^2Gx  +  2Fi/+C  =  0 

ARTICLE  PAGE 

109.  The  ellipse  defined .         .179 

110.  The  first  standard  equation  of  the  ellipse        ....     180 

111.  To  trace  the  ellipse  ^4-^=1 182 

112.  Intrinsic  property  of  the  ellipse.     Second  standard  equation  .     183 

113.  Every  equation  of  the  form 

Ax^  +  Bi/''-}-2Gx-\-2Fi/  ■}-  C  =  0 

represents  an  ellipse  whose  axes  are  parallel  to  the  coordi- 
nate axes,  if  A  and  B  have  the  same  sign     ....     186 

114.  Reduction  of  the  equation  of  an  ellipse  to  a  standard  form     .     189 

III.    The  Hyperbola 

Special  Equation  of  the  Second  Degree 

Ax'^  -  Bi/-^-]-2Gx  +  2Fy  +  C  =  0 

115.  The  hyperbola  defined 190 

116.  The  first  standard  form  of  the  equation  of  the  hyperbola       .     191 

117.  To  trace  the  hyperbola —- ^  =  1 193 

118.  Intrinsic  property  of  the  hyperbola.     Second  standard  equa- 

tion          195 

119.  Every  equation  of  the  form 

Ax'^  -\-By^  +  2Gx  +  2Fy  ■]-  C  =  0 
represents  an  hyperbola  whose  axes  are  parallel  to  the  coor- 
dinate axes,  if  A  and  B  have  unlike  signs    ....     197 

120.  Summary 199 

TV.    Tangents,  Normals,  Polars,  Diameters,  etc. 

121.  Introductory 200 

122.  Tangent  to  the  conic  Ax"^  +  By'^  +  2  Gx  ^2  Fy  +  C  =  0  in 

terms  of  the  coordinates  of  the  point  of  contact :  the  secant 
method 200 

123.  Normal  to  the  conic  Ax'^  +  By^  +  2  Gx  +  2  Fy  +  C  =  0,  at  a 

given  point     . 203 

124.  Equation  of  a  tangent,  and  of  a  normal,  that  pass  through  a 

given  point  which  is  not  on  the  conic    .....     205 


CONTENTS 


XV 


125.     Through  a  given  external  point  two  tangents  to  a  conic  can 

be  drawn 206 

^26.     Equation  of  a  chord  of  contact 207 

127.  Poles  and  polars 209 

128.  Fundamental  theorem 210 

129.  Diameter  of  a  conic  section 211 

130.  Equation  of  a  conic  that  passes  through  the  intersections  of 

two  given  conies 213 


V.   Polar  Equation  of  the  Conic  Sections 

131.  Polar  equation  of  the  conic       ..... 

132.  From  the  polar  equation  of  a  conic  to  trace  the  curve 


214 
215 


CHAPTER  IX 

The  Parabola  y'^  —  ^px 

133.  Review 219 

134.  Construction  of  the  parabola 220 

135.  The   equation   of  the  tangent  to  the  parabola  y'^  ^'^px  in 

terms  of  its  slope 221 

136.  The   equation   of  the  normal   to  the  parabola  y^  =  ^px  in 

terms  of  its  slope 222 

137.  Subtangent   and   subnormal.     Construction  of  tangent  and 

normal 222 . 

138.  Some  properties  of  the  parabola  which  involve  tangents  and 

normals  .         .         .         .         .         .         .         .         .         -         .  225 

139.  Diameters 230 

140.  Some  properties  of  the  parabola  involving  diameters       .         .  232 

141.  The  equation  of  a  parabola  referred  to  any  diameter  and 

the  tangent  at  its  extremity  as  axes  ....     233 


CHAPTER  X 

The   Ellipse  ^  +  ^  =  1 
a2       62 


142.  Review 237 

143.  The  equation  of   the  tangent  to  the  ellipse [-  ^  =  1   in 

a^      }p- 

terms  of  its  slope 238 

144.  The  sum  of  the  focal  distances  of  any  point  on  an  ellipse 

is  constant;  it  is  equal  to  the  major  axis  .         .         .     239 


xvi  CONTENTS 

ARTICLE  PAGE 

145.  Construction  of  the  ellipse 240 

146.  Auxiliary  circles.     Eccentric  angle 242 

147.  The  subtangent  and   subnormal.     Construction   of  tangent 

and  normal    ..........  244 

148.  The  tangent  and  normal  bisect  externally  and   internally, 

respectively,  the  angles  between  the  focal  radii  of  the  point 

of  contact       ..........  246 

149.  The  intersection  of  the  tangents  at  the  extremity  of  a  focal 

chord 247 

150.  The  locus  of  the  foot  of  the  perpendicular  from  a  focus  upon 

a  tangent  to  an  ellipse   ........  248 

151.  The  locus  of  the  intersection  of  two  perpendicular  tangents 

to  the  ellipse  . 249 

152.  Diameters 250 

153.  Conjugate  diameters 252 

154.  Given  an  extremity  of  a  diameter,  to  find  the  extremity  of 

its  conjugate  diameter 253 

155.  Properties  of  conjugate  diameters  of  the  ellipse       .         .         .  254 

156.  Equi-con jugate  diameters 257 

157.  Supplemental  chords 259 

158.  Equation  of  the  ellipse  referred  to  a  pair  of  conjugate  diam- 

eters          260 

159.  Ellipse  referred  to  conjugate  diameters ;  second  method          .  261 

CHAPTER  XI 

The  Hyperbola  - —  '^  =  1 

160.  Review 265 

161.  The  difference  between  the  focal  distances  of  any  point  on  an 

hyperbola  is  constant ;  it  is  equal  to  the  transverse  axis      .  266 

162.  Construction  of  the  hyperbola 267 

163.  The  tangent  and  normal  bisect  internally  and  externally  the 

angles  between  the  focal  radii  of  the  point  of  contact  .         .  268 

164.  Conjugate  hyperbolas 270 

165.  Asymptotes        .         .         .         .         .     - 272 

166.  Relation  between  conjugate  hyperbolas  and  their  asymptotes  275 

167.  Equilateral  or  rectangular  hyperbola 277 

168.  The  hyperbola  referred  to  its  asymptotes          ....  278 

169.  The  tangent  to  the  hyperbola  xy  =  c^ 280 

170.  Geometric  properties  of  the  hyperbola      .....  281 


CONTENTS 


XVll 


171.  Diameters 284 

172.  Properties  of  conjugate  diameters  of  the  hyperbola        .         .  285 

173.  Supplemental  chords 287 

174.  Equations  representing  an  hyperbola,  but  involving  only  one 

variable 288 

CHAPTER   XII 
General  Equation  of  the  Second  Degree 

Ax^  +  2  Hxy  +  Bf  +  2Gx-h2Fi/+  C=zO 

175.  General  equation  of  the  second  degree  in  two  variables  .         .  292 

176.  Illustrative  examples 294 

177.  Test  for  the  species  of  a  conic 297 

178.  Center  of  a  conic  section 298 

179.  Transformation  of  the  equation  of  a  conic  to  parallel  axes 

through  its  center ,  299 

180.  The  invariants  A  +  B  andH^  -  AB 301 

181.  To  reduce  to  its  simplest  standard  form  the  general  equation 

of  a  conic 303 

182.  Summary 306 

183.  The  equation  of  a  conic  through  given  points          .        .         .  307 


184.     Definitions 


CHAPTER   XIII 
Higher  Plane  Curves 


309 


I.   Algebraic  Curves 

185.  The  cissoid  of  Diodes 

186.  The  conchoid  of  Nicomedes 

187.  The  witch  of  Agnesi 

188.  The  lemniscate  of  Bernouilli 
189  a.  The  lima9on  of  Pascal 
189  6.  The  cardioid      . 
190.     The  Neilian,  or  semi-cubical  parabola 


II.    Transcendental  Curves 


191.  The  cycloid 

192.  The  hypocycloid 


309 
312 
314 
315 
318 
319 
320 


321 
323 


XVlll 


CONTENTS 


III.    Sjnrals 

AETICLE 

193.  Definition 

194.  The  spiral  of  Archimedes 

195.  The  reciprocal,  or  hyperbolic,  spiral 

196.  The  parabolic  spiral .        .        .        . 

197.  The  lituus  or  trumpet 

198.  The  logarithmic  spiral 


325 
325 
326 

328 
328 
329 


PART   II.  — SOLID   ANALYTIC   GEOMETRY 


CHAPTER  I 


199. 
200. 
201. 
202. 
203. 
204. 

205. 

206. 
207. 


Coordinate  Systems.     The  Point 

Introductory 331 

.  332 

.  333 

.  333 

.  334 


Rectangular  coordinates    ........ 

Polar  coordinates       ......... 

Relation  between  the  rectangular  and  polar  systems 

Direction  angles  :  direction  cosines  .         .         .        ... 

Distance  and  direction  from  one  point  to  another;  rectangu- 
lar coordinates        ......... 

The  point  which  divides  in  a  given  ratio  the  straight  line 
from  one  point  to  another 

Angle  between  two  radii  vectores.     Angle  between  two  lines 

Transformation  of  coordinates ;  rectangular  systems 


336 

337 
338 
339 


CHAPTER  II 

The  Locus  of  an  Equation.     Surfaces 

208.  Introductory 342 

209.  Equations   in   one   variable.     Planes  parallel   to   coordinate 

planes 343 

210.  Equations  in  two  variables.     Cylinders  perpendicular  to  coor- 

dinate planes 344 

211.  Equations  in  three  variables.     Surfaces 346 

212.  Curves.     Traces  of  surfaces 347 

213.  Surfaces  of  revolution        .        .        .    - 348 


CONTENTS  XIX 

CHAPTER  III 

Equations  of  the  First  Degree  Ax  -{-  Bij  -{■  Cz  +  D  =  0.     Planes 

AND  Straight  Lines 

I.    The  Plane 

ARTICLE  PAGE 

214.  Every  equation  of  the  first  degree  represents  a  plane     .         .  353 

215.  Equation  of  a  plane  through  three  given  points      .         .         .  354 

216.  The  intercept  equation  of  a  plane 354 

217.  The  normal  equation  of  a  plane       ......  355 

218.  Reduction  of  the  general  equation  of  first  degree  to  a  stand- 

ard form.    Determination  of  the  constants  a,  6,  c,  p,  a,  jS,  y    356 

219.  The  angle  between  two  planes.     Parallel  and  perpendicular 

planes 357 

220.  Distance  of  a  point  from  a  plane 359 

II.    The  Straight  Line 

221.  Two  equations  of  the  first  degree  represent  a  straight  line   .     359 

222.  Standard  forms  for  the  equations  of  a  straight  line 

^        (a)    The  straight  line  through  a  given  point  in  a  given 

direction 360 

(b)  The  straight  line  through  two  given  points  .         .        .     360 

(c)  The  straight  line  with  given  traces  on  the  coordinate 

planes 361 

223.  Reduction  of  the  general  equations  of  a  straight  line  to  a 

standard  form.     Determination  of  the  direction  angles  and 
traces 
I.   Third  standard  form  :  traces  ......     362 

II.   First  standard  form  :  direction  angles    ....     362 

224.  The  angle  between  two  lines ;  between  a  plane  and  a  line      .     363 


CHAPTER  IV 
Equations  of  the  Second  Degree.    Quadric  Surfaces 

225.  The  locus  of  an  equation  of  second  degi'ee      ....     367 

226.  Species  of  quadrics.     Simplified  equation  of  second  degree     .     368 

227.  Standard  forms  of  the  equation  of  a  quadric  ....     370 

2  2  2 

228.  The  ellipsoid :  equation  -^  +  ^  +  %  =  l  .        .        .         .371 

a^     0^      c^ 


XX 


COIiTENTS 


ARTICLE 

229.     The  un-parted  hyperboloid  :  equation  -2  +  ^ 


230.  The  bi-parted  hyperboloid:  equation  -^~jo  — 

231.  The  paraboloids  :  equation  ^  ±  ^  =  2 


~2 


X       y 
232.     The  cone:  equation  — „  +  t; 

o/'      0 


-.  =  0 


233.     The  hyperboloid  and  its  asymptotic  cone 


373 

375 
376 

378 
379 


APPENDIX 


Note  A. 
Note  B. 
Note  C. 
Note  D. 
Note  E. 
Note  F. 
Answers 
Index 


Historical  sketch 

Construction  of  any  conic           .... 
SjDecial  cases  of  the  conies          .... 
Every  section  of  a  cone  by  a  plane  is  a  conic    . 
Parabola  as  a  limiting  form  of  ellipse  or  hyperbola 
Confocal  conies 


381 
382 
383 
384 
387 
388 
391 
000 


AI^ALYTIC    GEOMETRY 


PART   I 


CHAPTER   I 
INTRODUCTION 

ALGEBRAIC  AND   TRIGONOMETRIC   CONCEPTIONS 

1.  Number.  A  number  is  most  simply  interpreted  as 
expressing  the  measurement  of  one  quantity  by  another 
quantity  of  the  same  kind  first  chosen  as  a  unit  of  measure ; 
it  is  positive,  or  +,  if  the  measuring  unit  is  taken  in  the 
same  sense  as  the  thing  measured;  and  negative,  or  — ,  if 
this  measuring  unit  is  taken  in  the  opposite  sense. 

I^.g.^  the  unit  dollar  may  be  regarded  as  a  dollar  of  assets, 
or  as  a  dollar  of  liabilities ;  if  it  is  regarded  as  a  dollar  of 
assets,  then  assets  measured  by  it  produce  positive  numbers, 
while  liabilities  measured  by  it  produce  negative  numbers. 

The  above  definition  is  consistent  with  the  one  usually 
given;  viz.  that  numbers  are  positive  or  negative  according 
as  they  are  greater  or  less  than  zero. 

If  tlie  operations  of  addition,  subtraction,  multiplication, 
division,  raising  to  integer  powers,  extracting  roots,  or  any 
combination  of  these  operations,  are  performed  upon  given 
numbers,  the  result  in  every  case  is  a  number  ;  it  is  imaginary 

TAN.  AX.  GEOJI.  —  1 


2  ANALYTIC  GEOMETRY  [Ch.  I. 

if  it  involves  in  any  way  whatever  an  indicated  even  root  of 
a  negative  number;  otherwise  it  is  real. 

Every  imaginary  number  may  be  reduced  to  the  form 
a  +  6  V—  1,  where  a  and  h  are  real,  and  b  =^0. 

2.    Constants  and  variables.     If  AB  and  AC  are  two  given 

straight  lines  making  an  angle  a  at 
the  point  A,  and  if  any  tAvo  points 
X  and  F,  on  these  lines,  respectively, 
^  ^    are  joined  by  a  straight  line,  then 

Area  of  triangle  AXY  =\'  AX-  A Y •  sin  a, 

i.e.^  A  =  ^  '  x- ^  '  sin  a^ 

where  x  is  the  length  of  AX,  y  is  the  length  of  A  Z,  and  A  is 
the  area  of  the  triangle. 

If  now  the  points  X  and  Y  are  moved  along  the  lines  AB 
and  AC  vn  any  way  whatever,  then  A,  x,  and  y  will  each  pass 
through  a  series  of  different  values,  —  they  are  variable  num- 
bers or  variables ;  while  \  and  sin  «  will  remain  unchanged, — 
they  are  constant  numbers  or  constants. 

It  is  to  be  remarked  that  \  has  the  same  value  wherever  it 
occurs,  —  it  is  an  absolute  constant;  while  «,  though  constant 
for  this  series  of  triangles,  may  have  a  different  constant 
value  for  another  series  of  triangles,  —  it  is  an  arbitrary 
constant. 

Because  x  and  y  may  separately  take  any  values  what- 
ever they  are  independent  variables;  while  A,  whose  value 
depends  upon  the  values  of  x  and  ?/,  is  a  dependent  variable. 

The  illustrations  just  given  may  serve  to  give  a  clearer 
conception  of  the  following  more  formal  definitions. 

An  absolute  constant  is  a  number  which  has  the  same  value 

2 

wherever  it  occurs;  such  are  the  numbers  2,  7,  -|,  6%   tt,  e 


1-3.  ]  INTR  OD  UCTION  3 

(where  ir  =  3.14159265..-,  approximately  -y-,  the  ratio  of  the 
circumference  of  a  circle  to  its  diameter;  and 

^  =  2.71828182...  =  1  +  ^  +  |-+  -|+  ..., 

approximately  -y-,  the  base  of  the  Naperian  system  of  loga- 
rithms). 

An  arbitrary  constant  is  a  number  which  retains  the  same 
value  throughout  the  investigation  of  a  given  problem,  but 
may  have  a  different  fixed  value  in  another  problem. 

An  independent  variable  is  a  number  that  may  take  any 
value  whatever  within  limits  prescribed  by  the  conditions  of 
the  problem  under  consideration. 

A  dependent  variable  is  a  number  that  depends  for  its 
value  upon  the  values  assumed  by  one  or  more  independent 
variables.* 

A  number  that  is  greater  than  any  assignable  number, 
however  great,  is  an  infinite  number;  one  that  varies  and 
becomes  and  remains  smaller  (numerically,  not  merely  alge- 
braically less)  than  any  assigned  number,  however  small,  is 
an  infinitesimal  number.     All  other  numbers  are  finite. 

3.  Functions.  A  number  so  related  to  one  or  more  other 
numbers  that  it  depends  upon  these  for  its  value,  and  takes 
in  general  a  definite  value,  or  a  finite  number  of  definite 
values,  when  each  of  these  other  numbers  takes  a  definite 
value,  is  a  function  of  these  other  numbers.  -E^.^.,  the  cir- 
cumference and  the  area  of  a  circle  are  functions  of  its  radius; 
the  distance  traveled  by  a  railway  train  is  a  function  of  its 
time  and  rate;  if  ?/  =  3  a;^  -f-  5iK  —  8,  then  ?/  is  a  function  of  x. 


*  All  these  kinds  of  numbers  will  be  met  and  better  illustrated  in  succeed- 
ing chapters  of  this  book.     E.g.^  see  Art.  65,  Note. 


4  ANALYTIC  GEOMETRY  [Ch.  1. 

4.  Identity,  equation,  and  root.  If  two  functions  involv- 
ing the  same  variables  are  equal  to  each  other  for  all  values 
of  those  variables  they  are  identically  equal.  Such  an 
equality  is  expressed  by  writing  the  sign  =  between  the 
two  functions,  and  the  exj)ression  so  formed  is  an  identity. 
If,  on  the  other  hand,  the  two  functions  are  equal  to  each 
other  only  for  particular  values  of  the  variables,  the  equality 
is  expressed  by  writing  the  sign  =  between  the  two  func- 
tions, and  the  expression  so  formed  is  an  equation.  The 
particular  values  for  which  the  two  functions  are  equal,  i.e., 
those  values  of  the  variables  which  satisfy  the  equation,  are 
the  roots  of  the  equation. 

E.g.,  {x  +  yy^=x^-  -\-2xy  +  y%     (x  +  a)(x  -  a)  +  a^  =  x% 

,  ,       3         .r^  —  a:  +  3 

and  X  H = 

x—1  x—1 

are  identities ;  while  3  x^  —  10  x  +  2  =  2x^  -  4:X  —  6,  or,  what  is  the  same 
thing,  a;2  -  6  X  +  8  =  0,  is  an  equation.  The  roots  of  this  equation  are 
the  numbers  2  and  4. 

Special  attention  is  called  to  the  fact  that  an  equation 
always  imposes  a  condition. 

E.g.,  x2  —  6x  +  8  =  0  if,  and  only  if ,  x  =  2  or  x  =  4.  So  also  the  equa- 
tion ax  -\- hy  -{■  c  =  ^  imposes  the  condition  that  x  shall  be  equal  to 

-by-c^ 
a 

5.  Functions  classified.  A  functional  relation  is  usually 
expressed  by  means  of  an  equation  involving  the  related 
numbers.  If  the  form  of  this  equation  is  such  that  one  of 
the  variables  is  expressed  directly  in  terms  of  the  others,  then 
that  variable  is  called  an  explicit  function  of  the  others;  if 
it  is  not  so  expressed,  it  is  an  implicit  function. 

E.g.,  the  equations  y  =  V o  —  x%  x^  -}-  y'^  =  5,  and  x  =  V5  —  y'^  express 
the  same  relation  between  x  and  y  ;  in  the  first  y  is  an  explicit  function 


4-6.]  INTttODtWnon  5 

of  X,  in  the  second  each  is  an  implicit  function  of  the  other,  while  in 
the  third  :*:  is  an  explicit  function  of  y. 

The  word  "function"  is,  for  brevity,  usually  represented 
by  a  single  letter,  such  as /,  jF,  (/>,  i|r,... ;  thus  y  =  <^(x)  means 
that  ^  is  a  function  of  the  independent  variable  x,  and  is  read 
"^  equals  the  (/)-f unction  of  a:";  so  also  z  =  F(ii,  v,  x) 
means  that  2  is  a  function  of  the  independent  variables  u,  v, 
and  X,  and  it  is  read,  "2  equals  the  ^-function  of  w,  v,  and  x.'' 

A  function  is  algebraic  if  it  involves,  so  far  as  the  inde- 
pendent variables  are  concerned,  only  a  finite  number  of  the 
operations  of  addition,  subtraction,  multiplication,  division, 
raising  to  integer  powers,  and  extracting  roots.  All  other 
functions  are  transcendental. 

E.q.,  2  x^  —  5  a:  —  17,    rrv  +  v^  —  7x,  and  ^-,   are  algebraic 

if  i  •>      i)       i)  X  ^  xy  —  1  y-  ° 

functions;  while  2^,  (f,  sin  a;,  tan~^2,  and  log  <  are  transcendental  func- 
tions. 

6.  Notation.  In  general,  absolute  constants  are  repre- 
sented by  the  Arabic  numerals,  while  arbitrary  constants  and 
variables  are  represented  by  letters.  A  few  absolute  con- 
stants are,  however,  by  general  consent,  represented  by  let- 
ters;  examples  of  such  constants  are  tt  and  e  (Art.  2). 
Variables  are  usually  represented  by  the  last  letters  of  the 
alphabet,  such  as  w,  v,  w^  x^  y^  z  \  while  the  first  letters, 
a^h^c^"'  are  reserved  to  represent  constants. 

Particular  fixed  values  from  among  those  that  a  variable 
may  assume  are  sometimes  in  question;  e.^.,  the  values, 
a:  =  2  and  2;  =  —  1,  for  which  the  function  a;^—  2:  —  2  vanishes; 
such  values  may  conveniently  be  denoted  b}^  affixing  a  sub- 
script to  the  letter  representing  the  variable.  Thus  x^^  x^,  ^31  *  *  * 
will  be  used  to  denote  particular  values  of  the  variable  x. 

Similarly,  variables  which  enter  a  problem  in  analogous 


6  ANALYTIC  GEOMETRY  [Ch.  I. 

ways  are  usually  denoted  by  a  single  letter  having  accents 
attached  to  it ;  thus  a;',  x'^ ^  x'",  •••  denote  variables  that  are 
similarly  involved  in  a  given  problem. 

Again,  each  of  the  two  equations,  ^=Bx^  —  4:x-]-10  and 
y  =  <^(a;),  asserts  that  y  is  a  function  of  x ;  but  while  the 
former  tells  precisely  how  y  depends  upon  x^  the  latter 
merely  asserts  that  there  is  such  a  dependence,  without 
giving  any  information  concerning  the  form  of  that  depend- 
ence. If  several  different  forms  of  functions  present  them- 
selves in  the  same  problem,  they  are  represented  by  different 
letters,  each  letter  representing  a  particular  form  for  that 
problem,  though  it  may  be  chosen  to  represent  an  entirely 
different  form  in  another  problem. 

E.g.,  if  the  form  of  cfi,  in  a  given  problem,  is  defined  by  the  equation 

3  x^  -  x^+  6 


<l>(x)  = 


then,  in  the  same  problem, 


2a,- +  1      ' 


<^(^0-^t~r^^-     <^a)  =  L     and     cf>(0)  =  5. 
2  y  +  1  o 


7.  Continuous  and  discontinuous  functions.  In  general  a 
function  takes  different  values  when  different  values  are 
assigned  to  its  independent  variable.  If  ?/  =  </>(^)?  then, 
for  X  =  a  and  x  =  d,  the  function  becomes  y-^  =  </>(<^)  and 
^2  =  <^(^),  and  ?/j  is  in  general  different  from  y^-  The  func- 
tion (j>(^x')  is  said  to  be  a  continuous  function  of  x  between 
x  =  a  and  x  =  b,  if,  while  x  is  made  to  pass  successively 
through  all  real  values  from  a  to  6,  y  remains  real  and  finite 
and  passes  corresj)ondingly  through  all  values  from  y^  to  y^' 

This  definition  may  be  more  precisely  stated,  thus :  If  a:^  and  ajg  are 
any  real  values  of  x  which  lie  between  the  values  a  and  b,  and  if  the  cor- 
responding values  of  y,  viz.  <f>(xj)  and  <j>(x.^,  are  real  and  finite;  and  if 


6-7.]  INTRODUCTION  i 

a  positive  number  rj  can  be  found,  such  that  by  tailing,  numerically, 

^l  —  -^2  *^  'Z' 

it  will  follow  that,  numerically, 

<f>(x^)  -  <f>(x2)  <  e, 

where  e  is  any  assigned  positive  number,  however  small ;  then  <f>(x)  is  a 
continuous  function  of  x  for  values  from  a  to  b. 

Or,  in  words :  y  h  a  continuous  function  of  x  for  all  values  of  x  in  the 
interval  from  a  to  b,  if,  by  taking  any  two  values  of  x  in  the  interval 
sufficiently  near  together,  the  difference  between  the  corresponding  values 
of  y  can  be  made  less  than  any  assigned  number,  however  small. 

A  discontinuous  function  is  one  that  does  not  fulfil  the 
conditions  for  continuity.  It  is,  however,  usually/  discon- 
tinuous for  only  a  limited  number  of  particular  values  of  its 
independent  variable,  while  between  these  values  it  is  con- 
tinuous. 

As  familiar  examples  of  continuous  functions  may  be 
mentioned :  the  length  of  a  solar  shadow ;  the  area  of  a 
cross-section  of  a  growing  tree,  or  of  a  growing  peach ;  the 
height  of  the  mercury  in  a  barometer  ;  the  temperature  of  a 
room  at  varying  distances  from  the  source  of  heat ;  and 
interest  as  a  function  of  time. 

So,  also,  y=3a;2-f4a;  +  l  is  a  continuous  function  of  x 
for  all  finite  values  of  x. 

For,  7/  remains  real  and  finite  so  long  as  x  remains  real  and 
finite,  and,  if  x-^  and  X2  be  any  two  finite  values  of  x  which 
differ  from  each  other  by  7;,  i.e.,  if  X2  =  x^±  7/,  then 

1^2  -  ?/i  =  3  2:22  +  4  3^2  +  1  -  (^  ^1^  +  ^  ^1  +  1)' 

=  3(2:1  ±  vY  4-  4(2^1  ±  7;)  +  1  -  (3:ri2  +  42:1  +  1), 

=  ±(62:1-^4  +  377)7;. 

Now  to  show  that  ?/  =  3  2:^  +  4  2;  +  1  is  continuous  for 
X  =  2^1,  it  only  remains  to  show  that,  by  taking  rj  sufficiently 


8  ANALYTIC   GEOMETRY  [Ch.  I. 

small,  ^.e.,  by  taking  x^  sufficiently  near  a^j,  7/^  can  be  made 
to  differ  from  ?/^  by  less  than  any  assigned  number  (e),  how- 
ever small.  But  this  is  evident;  for  y  may  be  taken  as  near 
zero  as  desired,  hence  the  factor  6  a;^  +  4  +  3  ?;  as  near  6  2:^  +  4 
as  desired,  and  the  product  therefore  as  near  zero  as  is  neces- 
sary to  be  less  than  e. 

On  the  other  hand,  if,  at  regular  intervals  of  time,  apples 
are  dropped  into  a  basket,  the  combined  weight  of  the  basket 
and  apples  will  increase  discontinuously ;  ^.e.,  their  total 
weight  is  a  discontinuous  function  of  the  time. 

EXERCISES 

1.  li  Ax  -]-  By  -\-C  =  0,  prove  that  ?/  is  a  continuous  function  of  x\ 
and  X,  of  y. 

2.  If  ,r2+  ?/2_  4  _  0^  prove  that  ?/  is  a  continuous  function  of  x,  when 
2>x>-2. 

3.  If  ^  -I-  ^_  =  1,  prove  that  ar  is  a  continuous  function  of  y,  when 
b>y>-b. 

4.  If ^ —  1  =  0.  is  a;  a  continuous  function  of  ?/ ? 

5.  If  61/  —  9  =  0,  is  5  a  continuous  function  of  tl 

6.  If  ifi  —  3  y  =  0,  is  i<  a  continuous  function  of  v  ?     Is  v  a  continu- 
ous function  of  u  ? 

7.  Show  that  all  functions  of  the  form 

OoX"  +  a^.r"-^  +  a2-2^«-2  +  •••  +  an-\X  +  ff„, 

where  CTq,  a^  a<)--' an  are  constants,  are  continuous  for  all  finite  values 
of  X. 

8.  If  " =  5^"\  show  that  y  is  discontinuous  for  x  =  1. 

y  ~  - 

1 

9.  Find  the  value  of  x  for  which  y,  —  c  ^'"'^  ~    ,  is  discontinuous. 


7-9.]  INTRODUCTION  9 

10.    Interest  on  money  loaned  is  calculated  by  the  formula 

I^  P-R'T. 
Is  the  interest  (/)  a  continuous  or  a  discontinuous  function  of  P? 
of /2?  of  r? 

8.  The  present  work  will  be  concerned  for  the  most  part 
with  algebraic  functions  involving  only  the  first  and  second 
powers  of  the  variable,  i.e.,  with  algebraic  equations  of  the 
first  and  second  degree.  A  review  is  therefore  given  of  the 
solution  and  theory  of  the  quadratic  equation,  presenting  in 
brief  the  most  important  results  which  will  be  needed  in  the 
Analytic  Geometry.  The  student  should  become  thoroughly 
familiar  with  this  theory,  as  well  as  with  the  review  of  the 
trigonometry  which  follows  it. 

9.  The  quadratic  equation.  Its  solution.  The  most  general 
equation  of  the  second  degree,  in  one  unknown  number,  may 
be  written  in  the  form 

ax^  -\-  bx  -\-  c  =  0,         .         .         .        (1) 

where  «,  5,  and  c  are  known  numbers.  This  equation  may 
be  solved  by  the  method  of  "  completing  the  square,"  which 
gives 

x^  -\--x  +  (--]=[--]  --,      .     .      .      (i) 
a  \2aJ       \2aJ       a 


h         .      Ir  b\^      c        .1 


i.e.,    x+^=±^^)  -^=±^yb^-^ac,     .     .     .   (3) 


2  a  ^\2aJ       a  2a 


b 


whence  x  = ±  — -  V^^  —  4  ac.       .       .       .        (4) 

2a      2a 

If  x-^  and  x^  are  used  to  denote  the  roots  of  eq.  (1),  they 
may  be  written 


X.  = ,  and  x^  = .   ...    (5) 

2a  2a 


10  ANALYTIC  GEOMETRY  [Ch.  I. 

The  nature  of  the  roots  (5)  depends  upon  the  number 
under  the  radical  sign,  i.e.,  upon  5^  —  4 «(?,  giving  three 
cases  to  be  considered,  viz.: 

if  P  —  4  ac  >  0,  then  the  roots  are  both  real  and  unequal, " 

a  P  —  4:  ac  =  0,  then  the  roots  are  both  real  and  equal,       '  (6) 

if  b^  —  4:  ac  <  0,  then  the  roots  are  both  imaginary. 

Thus  the  chat^acter  of  the  roots  of  a  given  quadratic  equa- 
tion may  be  determined  without  actually  solving  the  equation, 
by  merely  calculating  the  value  of  the  expression  6^  —  4  ac. 
This  important  expression  is  called  the  discriminant  of  the 
quadratic  equation ;  when  equated  to  zero  it  states  the  co7i- 
dition  that  must  hold  among  the  coefficients  if  the  equation 
has  equal  roots. 

EXERCISES 

1.  Show  which  of  the  following  equalities  are  identities : 

(1)  a-2-4.r  +  4  =  0;  (4)   {p  +  qy  =  p^  ^  ^3  +  3^,^  (p  +  ^). 

(2)  {s  +  t){s-t)  =  s^-t'^',        (5)  a;2  +  5 a;  +  6  =  (a;  +  3)(a;  +  2). 

a  +  p 

2.  Determine,  without  solving  the  equation,  the  nature  of  the  roots  of 

3a:2  +  8a;+l  =  0. 

Solution.  Since  6^  _  4  qc  =  64  —  12  =  52,  i.e.,  is  positive,  therefore 
the  roots  are  real  and  unequal ;  again,  since  a,  b,  and  c  are  all  positive, 
therefore  both  roots  are  negative  (cf.  eq.  (4),  Art.  9). 

3.  Without  solving  the  equation,   determine  the  character  of  the 
roots  of  Sx^  —  3x-\-l=0. 

4.  Given  the  equation  x^  —  dx  —  7n(x  +  2 x'^  +  4:)  =  5x^  +  3. 
Find  the  roots.     For  what  values  of  m  are  these  roots  equal  ? 

5.  Determine,  without  solving,  the  character  of  the  roots  of  the 
equations : 

(1)  5^2  _  2  2  +  5  =  0 ;     (2)  a;2  +  7  =  0 ;     (S)  dt'^  -  t  =  19. 


9-10.]  INTBODUCTION  11 

6.  Determine  the  values  of  m  for  which  the  following  equations  shall 
have  equal  roots : 

(1)  a;2  -  2a;(l  -f  3  m)  +  7  (3  +  2  m)=0) 

(2)  mx"^  +  2  z2  -  2  m  =  3  mx  -  9  a;  +  lO ; 

(3)  4x2  +  (l  +  m)^  +  1  =  0;         (4)  x2  +  (6x-  +  m)2  =  a2. 

7.  If  in  the  equation  2  ax  {ax  +  nc)  +  (n^  -  2)  c^  =  0,  a;  is  real,  show 
that  n  lies  between  —  2  and  +  2. 

8.  If   X   is   real   in  the  equation   ;: =  a,  show  that  a  lies 

between  1  and  —  ■^. 

9.  For  what  values  of  c  will  the  following  equations  have  equal  roots  ? 
(1)  3a:2  +  4x  +  c=0;     (2)   (mx  +  cy  =  ^lx;     (3)  4^2  +  9(2^;+ c)2  =  36. 

10.  Solve  the  equations  in  examples  2,  3,  and  5. 

11.  Solve  the  equations  : 

(1)  24-25a;2  =  -144;       (2)  3  a:  -  2  _  2  a:  +1  _^     12     ^  ^^ 
^^  '^^a;-2        a;  +  2a;2-4 

10.  Zero  and  infinite  roots.  In  tlie  following  pages  it  will 
sometimes  be  necessary  to  know  the  conditions  among  the 
coefficients  of  a  quadratic  equation  that  will  make  one  or 
both  of  its  roots  zero,  or  the  conditions  that  will  make  one 
or  both  of  the  roots  infinitely  large.  In  equations  (5)  of 
Art.  9,  x-^  and  x^^  i.e.  the  roots  of  ax^  -\- hx  -\-  e  =  0^  were 
found ;  and  it  is  at  once  seen  that 


—  h  -{-  V52  —  4  «c 
^1  = 7^ 


2a  -h--\/b^-4:ac      -h-^b^-4:ac 

and  that 


b-VW^4^c_  2c ...       (2) 

Equations  (1)  and  (2)  show  that : 

(1)  If  a  and  b  remain  unchanged  while  c  grows  smaller, 


12  ANALYTIC  GEOMETRY  [Ch.  I. 

then  x^  grows  smaller  and  x^  gro^YS  larger  ;  and  if  c  =  0,* 

then  X.  =  0,  while  x^  = 

i  "a 

(2)  \i  a  remains  unchanged  while  c  =  0  and  5  =  0,  then 
x^  =  0  and  a^g  =  0. 

(3)  If  5  and  c  remain  unchanged  while  a  =  0,  then  x-^= 

and  x^^  becomes  infinitely  large. 

(4)  If  c  remains  unchanged  while  «  =  0  and  5  =  0,  then 
both  x-^  and  x^  become  infinitely  large. 

(5)  If    a    and    c    remain   unchanged    while    5  =  0,    then 

x-,  =  \ and  x^  =  —\ 

The  student  should  translate  (1),  (2),  (3),  (4),  and  (5) 
into  more  general  terms  by  reading  "the  absolute  term 
approaches  zero  as  a  limit"  instead  of  "^  =  0,"  etc. 

11.   Properties  of  the  quadratic  equation.     By  adding  the 

two  roots  of 

a^  ^-hx-V  c^^  .        .        .        (1) 

and  also  multiplying  them  together,  the  relations 

X.  -\-  Xo  = and  x.x^  =  -      .      .      .      (2) 

a  ^      a 

are  obtained ;  or,  if  equation  (1)  is  written  with  the  coeffi- 
cient of  the  term  of  the  second  degree  reduced  to  unity,  as 

a^  -[-  px  -\-  q  =  0,        .         .        .         (3) 
these  relations  become 

^1  +  ^^2  —  ~  P  ^^^  ^1^2  —  9^  •  •  •  QK) 
Or,  expressed  in  words :  the  coefficient  of  the  term  of  the 
second   degree   being   unity,  the  coefficient  of  the  term  of 

*  The  sign  =  is  read  "  approaches  as  a  limit."  It  was  introduced  by  the 
late  Professor  Oliver  of  Cornell  University. 


10-12.]  INTRODUCTION  13 

the  first  degree  is  the  negative  of  the  sum  of  the  roots, 
while  tlie  term  free  from  x  is  the  product  of  the  roots. 

If,  therefore,  tlie  roots  of  a  quadratic  equation  are  not 
themselves  needed,  but  only  theu-  sum  or  product  is  de- 
sired, these  may  be  obtained  directly  from  the  given  equa- 
tion by  inspection. 

E.g.^  the  half  sum  of  the  roots  of  the  equation 

mV  +  2(hm  -  2  l)x  -\-h'^  =  0 

x^  +  x^  _  _  2 (5m  —  2  Z)  _  2  I  —  hm 
2  2m^  m^ 

Moreover,  if  x-^  and  x^  are  the  roots  of  the  equation 

a;^  +  pa:  +  5'  =  0,    • 

then  X  —  x-^  and  x  —  x^^  are  the  factors  of  its  first  member. 
For,  by  equation  (4)  above,  this  equation  may  be  written 

x^  -{-  px  -\-  q  =  x"^  —  (x-^  H-  x<^  X  -t-  x-^x^  =  0, 
and  x^  —  (x-^  -\-  x^  x  +  x-^x^  =  (2;  —  x-^) {x  —  x^^ 

hence  x^  +  px  +  g  =  (a;  —  x-^  (x  —  x^ . 

Conversely :  if  a  quadratic  function  can  be  separated  into 
two  factors  of  the  first  degree,  then  the  roots  can  be  imme- 
diately written  by  inspection. 

For,  if  x^  -\- px  -{-  q  =  (x  —  x-^(x  —  x^^  then  the  first  mem- 
ber will  vanish  if,  and  only  if ,  2:  —  a;-^  =  0  ov  x  —  x<^  =  0  ;  i.e. 
x^  -\-  px  -\-  q  =  0  if  X  =x-^^  or  x  =  Xg,  hence  x^  and  x^  are  the 
roots  of  the  equation  x^  -\- px  -{-  q  =  0  (cf.  Art.  4). 

12.   The  quadratic  equation  involving  two  unknowns.     One 

equation  involving  two  unknown  numbers  cannot  be  solved 
uniquely  for  the  values  of  those  numbers  which  satisfy  the 
equation ;  but  if  there  is  assigned  to  either  of  those  num- 


14  ANALYTIC  GEOMETRY  [Ch.  I. 

bers  a  definite  value,  then  at  least  one  definite  and  corre- 
sponding value  can  be  found  for  the  other,  so  that,  this  pair 
of  values  being  substituted  for  the  unknown  numbers,  the 
equation  will  be  satisfied.  In  this  way  an  infinite  number  of 
pairs  of  values,  that  will  satisfy  the  equation,  may  be  found. 
If,  however,  the  equation  is  homogeneous  in  the  two  un- 
knowns, i.e.,  of  the  form 

aa^  -f-  bx^  -\-  cy^  —  0, 

then  the  ratio  x  :  y  may  be  regarded  as  a  single  number,  and 
the  equation  has  properties  precisely  like  those  discussed 
in  Arts.  9,  10,  and  11. 

To  solve  a  system  consisting  of  two  or  more  independent 
simultaneous  equations,  involving  as  many  unknown  ele- 
ments, it  is  necessary  to  combine  the  equations  so  as  to 
eliminate  all  but  one  of  the  unknown  elements,  then  to  solve 
the  resulting  equation  for  that  one,  and,  by  means  of  the 
roots  thus  obtained,  find  the  entire  system  of  roots. 

EXERCISES 

1.  Given  the  equation  x^  +  3  a:  —  4  +  m  (3  x^  —  4)  —  2  mx^  =  0,  find  the 
sum  of  the  roots;  the  product  of  the  roots;  also  the  factors  of  the  first 
member. 

2.  Factor  the  following  expressions  : 

(1)  a;2-5a:  +  4;         (3)    mx^-3x-}-c;  (5)    3  ^o^ - 94 ?f; ^  - 64  ; 

(2)  x^  +  2x-8',         (4)    ax^  +  bxy  +  cf;       (6)    U-27y-18y^. 

3.  Without  first  solving  the  equation 

x^  —  3  X  —  m  (x  -\-  2  x'^ -{-  4:)  =  bx^  -\-  S 

find  the  sum,  and  the  product,  of  its  roots.  For  what  value  of  m  are  its 
roots  equal?  For  what  value  of  m  do  both  its  roots  become  infinitely- 
large  ?  If  all  the  terms  are  transposed  to  one  member,  what  are  the 
factors  of  that  member? 

4.  Without  first  solving,  determine  the  nature  of  the  roots  of  the 
equation  (jn  —  2)  (log  xy  —  (2m  +  3)  log  x  —  4:m  =  0.  [Regard  log  x  as 
the  unknown  element.] 


12-13.]  INTRODUCTION  15 

For  what  values  of  m  are  the  roots  equal?  Real?  One  infinitely- 
great?  Both  infinitely  gTeat?  One  zero?  Find  the  factors  of  the  first 
member  of  the  equation. 

5.  Find  five  pairs  of  numbers  that  satisfy  the  equation  : 

(1)  x  +  3y-7=0;         (3)   y^  =  \Qx; 

(2)  a;2  +  2/2  =  4:;  (4)    3x  +  6x?/-83/2  +  3  a:2  =  0. 

6.  Without  solving,  determine  the  nature  of  the  roots  of  the  equation  : 

9x2+  \2xy  +  42/2  =  0,  3^2  -  wy  +  19^2  =  0. 

7.  Solve  the  following  pairs  of  simultaneous  equations  : 

(1)  3a:- 5y +  2  =  0,  and  2a; +  7?/ -4  =  0; 

(2)  5y  +  22  +  3  =0,  and7y  +  4^  +  2  =  0; 

(3)  y  =  3  a;  +  c  =  0,  and  y^  =  ^  x', 

(4)  a:2  +  2/2  =  5^  and  y^  =  Qx; 

(5)  h^x^  +  a^y'^  =  a^W;  and  y  =  ax  -\-  h', 

(6)  £--f.L  =  l,  and  --^  =  1. 
^  Me      9  16      9 

8.  Determine  those  values  of  h  for  which  each  of  the  following  pairs 
of  equations  will  be  satisfied  by  two  equal  values  of  y : 

(1)    {a;2  +  2/2  =  a2,  y  =  Qx  +  &};  (2)    {y  =  mx  +b,  y^  =  4a:}; 

(3)  {82/  +  2  a:  =  &,  6x2  +  ^2  ^  12}. 

9.  Determine,  for  the  pairs  of  equations  in  Ex.  8,  those  values  of  h 
which  will  give  equal  values  of  a:. 

TRIGONOMETRIC   CONCEPTIONS   AND  FORMULAS 

13.  Directed  lines.  Angles.  A  line  is  said  to  be  directed 
when  a  distinction  is  made  between  the  segment  from  any 
point  A  of  the  line  to  another  point  By  and  the  opposite  seg- 
ment from  B  to  A.  One  of  these  directions  is  chosen  as 
positive,  or  +,  and  the  opposite  direction  is  then  negative 
or  — . 

The  angle  formed  by  two  intersecting  directed  straight 
lines  is  that  relation  between  the  positions  of  the  two  lines 
which  is  expressed  by  the  amount  of  rotation  about  their 
point  of   intersection   necessary  to  bring  the   positive  end 


16  ANALYTIC  GEOMETRY  [Ch-  I. 

of  the  initial  side  into  coincidence  with  the  positive  end 
of  the  terminal  side.  The  point  in  wliich  the  lines  in- 
tersect is  called  the  vertex  of  the  angle.  The  angle  is 
positive^  or  4- ,  if  the  rotation '  from  the  initial  to  the  ter- 
minal side  is  in  counter-elockwise  direction ;  the  angle  is 
7iegative,  or   — ,  if  the  rotation  is  clockwise. 

The  angle  formed  by  two  directed  straight  lines  in  space, 
which  do  not  meet,  is  equal  to  the  angle  between  two  inter- 
secting lines,  which  are  respectively  parallel  to  the  given 
lines. 

For  the  measurement  of  angles  there  are  two  absolute 
units  : 

(1)  The  angular  magnitude  about  a  point  in  a  plane,  i.e., 
a  complete  revolution.  One  fourth  of  a  complete  revolution 
is  called  a  right  angle,  g^^  of  a  right  angle  is  a  degree  (1°), 
g^Q-  of  a  degree  is  a  minute  (1'),  and  ^^  of  a  minute  is  a 
second  (!"}  ; 

(2)  the  angle   whose   subtending  circular   arc   is  equal   in 

length   to   the    radius    of  that   arc;    this   angle   is   called   a 

radian  )T^^j  ;  it  is  independent  of  the  length  of  the  radius. 

o-        circumference     semi-circumference  .,  r?  n         ,i    , 

Since = : =  TT,  it  follows  that 

diameter  radius 

the  angle  formed  by  a  half  rotation,  i.e.,  180°,  is  ir  radians; 

i.e.,  180°  =  TT^'^^  =  (-yj     approximately; 

also         l^'')  =  i?^  =  57°  17'  44.8"  approximately. 

TT 

(r) 


A  right  angle  is  90°  or  f  —  j 


When  there  is  no  danger  of  being  misunderstood,  the  index 


TT  -,.  .  .,  ,  -1  TT 


(r)  is  omitted,  and  —   radians  is  written  simply  as  — ,  and 

...  if" 


13-14.] 


INTRODUCTION 


17 


14.  Trigonometric  ratios.  If  from  any  point  P  in  the  ter- 
minal side  of  an  angle  ^,  at  a  distance  r  from  the  vertex,  a 
perpendicular  MP  is  drawn  to  the  initial  side  meeting  it  in 


X  M 

Fig.  3.^ 


M        X        V   ^ 
Fig.  2v^ 


ilt/,  and  if  MP  be  represented  by  y  and  VM  by  x^  then,  by 
general  agreement,  y  is  +  if  MP  makes  a  positive  right 
angle  with  the  initial  line,  and  —  if  this  right  angle  is 
negative  ;  similarly,  a;  is  +  if  VM  extends  in  the  positive 
direction  of  the  initial  line,  and  —  if  it  extends  in  the 
opposite  direction. 

The  three  numbers  r,  x^  and  y  form  with  each  other  six 
ratios ;  these  ratios,  moreover,  depend  for  their  value  solely 
upon  the  size  of  the  angle  ^,  and  not  at  all  upon  the  value  of 
r.  These  six  ratios  are  known  as  the  trigonometric  ratios  or 
functions  of  the  angle  ^,  and  are  named  as  follows  : 


sine 


e  = 


y 


tangent  0  = 


secant  6  = 


X 


X  X 

cosine  0  =  -.        cotansfent  0  =-. 
r  ^  y 


cosecant  6  = 


1/ 


The  abbreviated  symbols  for  these  functions  are  sin  0, 
cos  0,  tan  6,  cot  6,  sec  ^,  and  esc  ^,  respectively.  The  func- 
tions are  not  all  independent,  but  are  connected  by  the  fol- 
lowing relations  : 


(1)  sin  0  •  CSC  =  1, 

(2)  cos  ^  •  sec  (9  =  1, 

(3)  tan  ^  .  cot  ^  =  1, 

(4)  tan  0  =  sin  6  :  cos  ^, 

TAN.   AN.  GEOM. 2 


(5)  cot  0  =  cos  6  :  sin  ^, 

(6)  sin2  0  -f-  cos2  (9  =  1, 

(7)  tan2  ^  +  1  =  sec2  6>, 

(8)  cot2  6  +  1  =  csc2  6. 


18  ANALYTIC  GEOMETRY  [Ch.  L 

By  means  of  these  eight  relations  all  the  trigonometric 
functions  of  any  angle  may  be  expressed  in  terms  of  any 
given  function.  -^.^■,  suppose  the  sine  of  an  angle  is  given, 
and  the  tangent  of  this  angle,  in  terms  of  the  sine,  is  wanted: 

by  (4),  tan  0  =  ^, 

-^  ^  ^  cos^ 

hence  tan  6  = 


and  by  (6),  cos  0  =  Vl  —  sin^  0, 

sill  6 


Vl-sin2(9 

If  the  numerical  value  of  sin^  is  given,  this  last  formula 

gives   the   corresponding  numerical  value  of  tan^;  e.g.,  if 

sin  ^  =  #,  then                               _3  q 

^  tan  ^  = s =  ±  T  • 

15.  Functions  of  related  angles.  Based  upon  the  defini- 
tions of  the  trigonometric  functions  the  following  relations 
are  readily  established. 

If  6  is  any  plane  angle,  then* 

(1)  sin  (  —  ^)  =  —  sin  0,  cos  (  —  ^)  =  +  cos  6, 
tan  (  —  ^)  =  —  tan  6,  esc  (  —  ^)  =  —  esc  6, 
sec  (  —  ^)  =  -f-  sec  6,           cot  (  —  ^)  =  —  cot  6 ; 

(2)  sin  (tt  ±  ^)  =  T  sin  6,  cos  (tt  ±  ^)  =  —  cos  0, 
tan  (tt  ±0)=  ±  tan  6,  esc  (tt  ±  ^)  =  T  esc  0, 
sec  (tt  ±6}=  —  sec  6,  cot  (tt  ±  ^)  =  ±  cot  ^ ; 

(3)  sin  fE±e^=  +  cos  6^,  cos/|  ±0\=T  sin  ^, 
tan^l^  T  6>')=  T  cot  0,  ^^^(^  ±o)=  +  sec  0, 
secf'^TO)^  T  CSC (9,  cot^^  ±0)=t  tan^. 

*  The  student  should  thoroughly  familiarize  himself  with  these  formulas, 
and  those  of  Art.  16,  as  well  as  with  the  derivation  of  each. 


14-16.]  INTRODUCTION  19 

16.   Other  important  formulas.     If  ^^  and  6^  ^^^  ^^J  two 
plane  angles,  then 

sin  (^^  ±  ^2)  =  sin  6^  cos  ^2  i  c^s  ^^  sin  0^, 

cos  (^j  ±  ^2)  =  cos  0-^  cos  ^2  -^  sin  ^^  sin  ^g^ 

4.      ra  ^  a  \       "tan  ^^  ±  tan  6^ 
tan  (^.  ±  a'o)  =  z ^-7^ ^• 

^  ^       ^^      1  T  tan  6>i  tan  ^2 
If  6  is  any  plane  angle,  then 

sin  2  ^  =  2  sin  0  cos  ^, 

cos  2  ^  =  cos2  6^  -  sin2  6'  =  1  -  2  ^\^2  0^2  cos2  ^  -  1, 

J.      oa         2  tan  ^ 
tan  26  = -— ^ 

1  -  tan2  e 
sin  -  =  Vj(l  —  cos  ^), 

Li 

COS  -  =  V^(l  +  cos  ^), 


Vi 


.      9  _  ^\\  —  cos  ^      1  —  cos  ^         sin  0 


2       ^  1  +  cos  ^         sin  ^  1  4-  cos  ^ 

If  a^  5,  and  <?  are  the  sides  of  a  triangle  lying  respectively 
opposite  the  angles  A^  B^  and  (7,  and  if  A  is  the  area  of  this 
triangle,  then 

^2  =  52  +  c2  _  2  5tf  cos  A^    and    L  =  ^hc  sin  A. 

EXERCISES 

1.  Express  in  radians  the  angles  : 

15°;    60°;    135°;    -252°;   f  rt.  angle ;    10°10'10";    88°2';     (37r)°. 

2.  Express  in  degrees,  minutes,  and  seconds,  the  angles : 

(f  )"* '   {tT  '   (i)'"  '   (1)"* '    To"^^  revolution  ;   5  rt.  angle. 

3.  Find  the  values  of  the  other  trigonometric  functions,  given : 

(1)    tan  0  =  3]      (2)    sec  a;  =  -  V2 ;      (3)    cos  <^  =  -^ ;       (4)    sin  ^  =  f ; 
(5)  cotil/  =  };  and  (6)  esc  w  =  -  2.  Y^ 


20 


ANALYTIC  GEOMETRY 


[Ch.  I. 


Solution  of  (1).  If  tan  0  =  3,  then  substituting  this  value  in  (3)  of 
Art.  14,  gives  cotO  =  i',  substituting  these  values  in  (7)  and  (8)  of  the 
same  article  gives  the  values  of  sec  0  and  of  esc  0;  and  substituting  those 
values  in  (1)  and  (2)  gives  sin  6  and  cos  (9. 

Another  method:     Construct  a  right  triangle  ABC  with  the  sides 
AB  =  1  and  BC  =  3,  then  ZBAC  is  an  angle  whose  tangent  is  3.     If 
AB  =  1  and  BC  =  S,  then  AC  =  vTo,  and  the  other  func- 
C  tions  of  the  angle  BA  C  are  at  once  seen  to  be : 


sin  0  = 


3 


cos  6 


1 


CSC  6 


10 


VlO  VlO  ^ 

sec  0  =  VlO,  and  cot  6  =  ^. 

Either  of  these  methods  may  be  employed  to  solve  the 
other  parts  of  this  example ;  the  second  method  is  usually 
to  be  preferred. 

4.   By  means  of  a  right  triangle,  with  appropriate  acute 
Fig  3    ^     angles,  find   the   numerical   values   of   the   trigonometric 
ratios  of  the  following  angles  : 

30°;  45°;  60°;  90°;  135°;  and  -45°. 

5.   Express  the  following  functions  in  terms  of  functions  of  positive 
angles  less  than  90°  : 

tan  3500° ;     -  esc  290° ;    sin  (  -  369°) ;     -  cos  ^^ ;   and   cot  (  -  1215°). 


6.    Solve  the  following  equations  : 

(1)  sin  ^  =  -  cos  210° ;    (2)  cos  ^  =  sin  2  ^ ;    (3) 

and  (4)    (sec^a:  —  l)(csc2a;  +  1)  = 


cos  X 


sin  X  cot^  X 


=  V3; 


7.   In  the  following  identities  transform  the  first  member  into   the 
second : 


(1) 


tan  0  -  cot 


-1; 


(2) 


sec  X  +  esc  X      1  +  cot  x 


tan  0  +  cot  0 ~  CSC-  6        '  ^  ''  sec  x  —  esc  x~  1  —  cot  x 

(3)  CSC  X  (sec  X  —  1)—  cot  a;  (1  —  cos  x)  =  tan  x  —  sin  x ; 

(4)  (2  r  sin  a  cos  a)^  +  r^  (cos^  a  —  sin^  a)^  ^  r^ ; 

(5)  (cos  a  cos  h  +  sin  a  sin  by  -\-  (sin  a  cos  h  —  cos  a  sin  by  =  l;  and 

(6)  (r  cos  <^)2  +  (r  sin  <^  cos  Oy  +  (r  sin  <^  sin  Oy^=l. 


16-17.] 


INTROnUCTtON 


21 


17.  Orthogonal  projection.  The  orthogonal  projection*  of 
a  point  upon  a  line  is  the  foot  of  the  perpendicular  from 
the  point  to  the  line.  In  the  figure,  M  is  the  projection 
of  P  upon  AB.     The  projection  of   a   segment  P§  of   a 


Q 


Fig.  4.- 


T^.'^^ 

^ 

H 
B 

A 

.-'< 

M  , 

N 

Fig.  iP- 


line  upon  another  line  AB,  is  that  part  of  the  second  line 
extending  from  the  projection  of  the  initial  point  of  the  seg- 
ment to  the  projection  of  the  terminal  point  of  the  segment. 
Thus  MN  is  the  projection  of  FQ  upon  AB,  and  NM  is  the 
projection  of  QP  upon  AB. 

The  length  of  the  projection  can  easily  be  expressed  in 
terms  of  the  length  of  the  segment  and  the  angle  which  it 
makes  with  the  line  upon  which  the  segment  is  projected  ;  for 

MJSr    PH 

= =  COS  «, 

PQ     PQ 

,',   MN=PQ  -  cosa; 

i.e.,  the  projection  of  a  segment  of  a  line  upon  another  line 
is  equal  to  the  product  of  its  length  hy  the  cosine  of  the  angle 
which  it  makes  ivith  that  other  line. 

A  line  made  up  of  parts  P$,  QR,  RS,  •••  (Fig.  ba,  55),  which 
are  straight  lines  having  different  directions,  is  a  broken  line  ; 
and  the  projection  of  a  broken  line  upon  any  line  is  the 
algebraic  sum  of  the  projections  of  its  parts  upon  the  same 


*  Hereafter,  unless  otherwise  stated,  projection  will  be  understood  to 
mean  orthogonal  projection. 


22 


AJS-ALTTIC  GEOMETBT 


[Ch.  I. 


line.     Thus  the  projection  of  PQRST  upon  AB  is  the  pro- 
jection of  PQ  -f-  the  projection  of  QR  +  •••,  upon  AB \  i.e., 

proj.  PQRST  upon  AB  =  MN  +  NK  +  KL  +  LH  =  MH\ 


Fig.  5.^ 


but  MHis  the  projection  of  the  straight  line  P2^  which  joins 
the  first  initial  to  last  terminal  point  of  the  broken  line.  In 
the  same  way  it  may  be  shown  that  the  projection  of  any 
broken  line  upon  a  straight  line  equals  the  projection,  upon 
the  same  straight  line,  of  the  straight  line  which  joins  the 
extremities  of  the  broken  line.  It  follows,  therefore,  that 
the  projection  of  the  perimeter  of  any  closed  j)olygon  uj)on 
any  given  line  is  zero. 

If  6>i,  l^g,  6'3,  ^4,  and  6^  be  the  angles  that  PQ,  QR,  RS, 
ST,  and  PT  respectively  make  with  the  line  AB,  then  the 
projection  of  the  broken  line  upon  AB  may  also  be  expressed 
thus : 

proj.  PQRST  upon  AB  =  MN+  NK^  KL  +  LH=  MH 
=  PQ  cos  (9i  +  QR  cos  6^  +  i^^S^cos  6^  +  STcos  0^ 
=  PT  cos  e^. 

The  projections  of  two  parallel  segments  of  equal  length 
upon  any  given  line  in  space  are  equal.  It  therefore  fol- 
lows that : 

(1)  The  projection  of  a  segment  of  a  line  upon  any  straight 


17.]  INTRODUCTION  23 

line  in  space  equals  the  product  of  its  length  by  the  cosine 
of  the  angle  between  the  two  lines. 

(2)  The  projection  of  any  broken  line  in  space  upon  any 
straight  line  equals  the  projection,  upon  the  same  line,  of 
the  straight  line  which  joins  the  extremities  of  the  broken 
line. 

EXERCISES 

1.  Two  lines  of  lengths  3  and  7  respectively  meet  at  an  angle  -;  find 
the  projection  of  each  upon  the  other. 

2.  The  center  of  an  equilateral  triangle,  of  side  5,  is  joined  by  a 
straight  line  to  a  vertex ;  find  the  projection  of  this  joining  line  upon 
each  side  of  the  triangle. 

3.  A  rectangle  has  its  sides  respectively  4  and  6 ;  find  their  projec- 
tions upon  a  diagonal. 

4.  Find  the  length  of  the  projection  of  each  edge  of  a  cube  upon 
a  chosen  diagonal. 

5.  A  given  line  AB  makes  an  angle  of  30°  with  the  line  MN,  and 
BC  is  perpendicular  to  AB  and  of  length  15;  find  the  projection  of 
BC  upon  MN. 

Solve  this  problem  if  the  given  angle  be  a  instead  of  30°. 

6.  Two  lines  in  space,  of  length  a  and  b  respectively,  make  an  angle 
Q)  with  each  other ;  find  the  projection  of  b  upon  a  line  that  is  perpen- 
dicular to  a. 

7.  Project  the  perimeter  of  a  square  upon  one  of  its  diagonals. 


CHAPTER   II 

GEOMETRIC  CONCEPTIONS.     THE   POINT 

I.     COORDINATE    SYSTEMS 

18.  Coordinates  of  a  point.  Position,  like  magnitude,  is 
relative,  and  can  be  given  for  a  geometric  figure  only  by 
reference  to  some  fixed  geometric  figures  (planes,  lines,  or 
points)  which  are  regarded  as  known,  just  as  magnitude 
can  be  given  only  by  reference  to  some  standard  magni- 
tudes which  are  taken  as  units  of  measurement.  The  posi- 
tion of  the  city  of  New  York,  for  example,  when  given  by  its 
latitude  and  longitude,  is  referred  to  the  equator  and  the 
meridian  of  Greenwich,  ^-  the  position  of  these  two  lines 
being  known,  that  of  New  York  is  also  known.  So  also 
the  position  of  Baltimore  may  be  given  by  its  distance  and 
direction  from  Washington ;  while  a  particular  point  in  a 
room  may  be  located  by  its  distances  from  the  floor  and 
two  adjacent  walls. 

If,  as  in  the  last  illustration,  a  point  is  to  be  fixed  in  space, 
then  three  magnitudes  must  be  known,  referring  to  three  fixed 
positions.  If,  on  the  other  hand,  the  point  is  on  a  known 
surface,  as  New  York  or  Baltimore  on  the  surface  of  the 
earth,  then  only  tioo  magnitudes  need  be  known,  referring  to 
two  fixed  positions  on  that  surface ;  while  if  the  point  is  on 
a  known  line,  onl}^  one  magnitude,  referring  to  one  fixed 
position  on  that  line,  is  needed  to  fix  its  position. 

These  various  magnitudes  which  serve  to  fix  the  position 

24 


Ch.  II.  18-20.]  GEOMETRIC  CONCEPTIONS  25 

of  a  point,  —  in  space,  on  a  surface,  or  on  a  line, —  are  called 
the  coordinates  of  the  point. 

19.  Analytic  Geometry.  Coordinates  may  be  represented 
by  algebraic  numbers ;  the  relations  of  the  various  points, 
and  tlie  properties  of  the  various  geometric  figures  which  are 
formed  by  those  points,  can  be  studied  through  the  corre- 
sponding relations  of  these  algebraic  numbers,  or  coordinates, 
expressed  in  the  form  of  algebraic  equations.  This  fact  is 
the  basis  of  analytic,  or  algebraic,  geometry,  the  main  object 
of  which  is  the  study  of  geometric  properties  by  algebraic 
methods. 

Analytic  geometry  may  be  conveniently  divided  into  two 
parts :  Plane  Analytic  Geometry,  which  treats  only  of  figures 
in  a  given  plane  surface  ;  and  Solid  Analytic  Geometry,  which 
treats  of  space  figures,  and  includes  Plane  Analytic  G-eometry 
as  a  special  case.  The  plane  analytic  geometry,  being  the 
simpler,  will  be  studied  first,  in  Part  I  of  tliis  book,  and 
Part  II  will  be  devoted  to  the  study  of  the  solid  analytic 
geometry.  In  this  first  part  of  the  subject  it  will  therefore 
be  understood  tliat  the  work  is  restricted  to  a  given  plane 
surface. 

Two  systems  of  coordinates  will  be  used,  the  Cartesian 
and  the  Polar.     They  are  explained  in  the  next  few  articles. 

20.  Positive  and  negative  coordinates.     If  a  point  lies  in  a 
given  directed  straiglit  line,  its  position  with  reference  to  a 
fixed  point  of  that  line  is  com- 
pletely determined  by  one  coor-      x'      P i P        x 

dinate.     E.g.,  let  X' OX  he  a  L--3----l-^-3--^ 

^   '  Fig.  6. 

given    directed    straight    line, 

and  let  distances  from  0  toward  X  be  regarded  as  positive, 

then  distances  from  0  toward  X'  are  negative.     A  point  P, 


«# 


26 


ANALYTIC  GEOMETRY 


[Ch.  IL 


in  this  line  and  3  units  from  0  toward  X  may  be  designated 
by  +  3,  where  the  sign  +  gives  the  direction  of  the  point, 
and  the  number  3  its  distance,  from  0.  Under  tliese  circum- 
stances the  point  P'  lying  3  units  on  the  other  side  of  0 
would  be  designated  by  —  3. 

In  the  same  way  there  corresponds  to  every  real  number, 
positive  or  negative,  a  definite  point  of  this  directed  straight 
line;  the  numbers  are  called  the  coordinates  of  the  points; 
and  0,  from  which  the  distances  are  measured,  is  called  the 
origin  of  coordinates. 


21.  Cartesian  coordinates  of  points  in  a  plane.  Suppose 
two  directed  straight  lines  X' OX  and  Y' OY  are  given, 
fixed  in  the  plane  and  intersecting  in  the  point  0,  These 
two  given  lines  are  called  the  coordinate  axes,  X'  OX  being 
the  a;-axis,  and  Y'OY  being  the  ?/-axis  ;  their  point  of  inter- 
section 0  is  the  origin  of  coordinates.     Any  other  two  lines, 

parallel  respectively  to  these  fixed 
lines,  and  at  known  distances  from 
them,  will  intersect  in  one  and  but 
one  point  P,  whose  position  is  thus 
definitely  fixed.  If  these  lines 
through  P  meet  the  axes  in  M  and 
L  respectively,  then  the  directed 
distances  LP  and  MP^  measured 
parallel  respectively  to  the  axes^  are  the  Cartesian  coordinates 
of  the  point  P.  The  distance  LP^  or  its  equal  OM,  is  the 
abscissa  of  P,  and  is  usually  represented  by  x^  while  JfP,  or 
its  equal  OL,  is  the  ordinate  of  P,  and  is  usually  represented 
by  y.  The  point  P  is  designated  by  the  symbol  (a:,  y), — often 
written  P  =  (x^y')^  —  the  abscissa  always  being  written  first, 
then   a  comma,  then  the   ordinate,  and  both  letters  being 


Fig.  7.— 


Y 

II. 

L 

I. 

0 

!    X 

M        ' 

III. 

IV. 

20-22.]  GEOMETRIC  CONCEPTIONS  27 

inclosed  in  a  parenthesis.  Thus  the  point  (4,  5)  is  the 
point  for  which  OM  =  4  and  MP  =  5  ;  while  the  point 
(-  3,  2)  has   0M=  -  3  and  MP  =  2. 

22.  Rectangular  coordinates.  The  simplest  and  most  com- 
mon form  of  Cartesian  coordinate  axes  is  that  in  which  the 
angle  XOY  is  a  positive  right 
angle ;  the  abscissa  (a:)  of  a 
point  is,  in  this  case,  its  perpen- 
dicular distance  from  the  y-axis, 
and  its  ordinate  (^)  is  its  perpen- 
dicular distance  from  the  a;-axis. 
This  way  of  locating  the  points 
of  a  plane  is  known  as  the  rec- 
tangular system  of  coordinates.  fig.7.- 
The  axes  divide  the  entire  plane  into  four  parts  called  quad- 
rants, which  are  usually  designated  as  first  (I),  second  (11), 
third  (III),  and  fourth  (IV),  in  the  order  of  rotation  from 
the  positive  end  of  the  2:-axis  toward  the  positive  end  of  the 
y-axis,  as  indicated  in  the  accompanying  figure. 

These  quadrants  are  distinguished  by  the  signs  of  the 
coordinates  of  the  points  lying  within  them,  thus  : 

in  quadrant  I  the  abscissa  (x^  is  -H,  the  ordinate  (?/)  is  + 
in  quadrant  II  the  abscissa  (x^  is  — ,  the  ordinate  (z/)  is  + 
in  quadrant  III  the  abscissa  (a;)  is  — ,  the  ordinate  (?/)  is  — 
in  quadrant  IV  the  abscissa  (2:)  is  H-,  the  ordinate  (^)  is  — . 

Four  points  having  numerically  the  same  coordinates,  but 
lying  one  in  each  quadrant,  are  symmetrical  in  pairs  Avith 
regard  to  the  origin,  even  though  the  axes  are  not  at  right 
angles ;  if,  however,  the  axes  are  rectangular,  then  these 
points  are  symmetrical  in  pairs,  not  merely  with  regard  to 
the  origin  as  before,  but  also  with  regard  to  the  axes,  and 


28  ANALYTIC  GEOMETRY  [Ch,  II. 

they  are  severally  equidistant  from  the  origin.  Because  of 
this  greater  symmetry  rectangular  coordinates  have  many 
advantages  over  an  oblique  system. 

In  the  folloiuing  pages  rectangular  coordinates  will  always 
be  understood  unless  the  contrary  is  expressly  stated, 

EXERCISES 

1.  Plot  accurately  the  pomts :  (1,  7),  (-4,  -5)  *  (0,  3),  and  ("3,  0). 

2.  Plot  accurately,  as  vertices  of  a  triangle,  the  points  :  (1,  3),  (2,  7), 
and  (~4,  ~4).  Find  by  measurement  the  lengths  of  the  sides,  and  the 
coordinates  of  the  middle  point  of  each  side. 

3.  Construct  the  two  lines  passing  through  the  points  (2,  -7)  and 
(-2,  7),  and  (2,  7)  and  ("2,  -7),  respectively.  What  is  their  point  of 
intersection  ?     Find  the  coordinates  of  the  middle  point  of  each  line. 

4.  If  the  ordinate  of  a  point  is  0,  where  is  the  point?  if  its  abscissa 
is  0?  if  its  abscissa  is  equal  to  its  ordinate?  if  its  abscissa  and  ordinate 
are  numerically  equal  but  of  opposite  signs  ? 

5.  Express  each  of  the  conditions  of  Ex.  4  by  means  of  an  equation. 

6.  The  base  of  an  equilateral  triangle,  whose  side  is  5  inches,  coincides 
with  the  X-axis ;  its  ndddle  point  is  at  the  origin  ;  what  are  the  coordinates 
of  the  vertices?  If  the  axes  are  chosen  so  as  to  coincide  with  two  sides  of 
this  triangle,  respectively,  what  are  the  coordinates  of  the  vertices? 

7.  A  square  whose  side  is  5  inches  has  its  diagonals  lying  upon  the 
coordinate  axes;  find  the  coordinates  of  its  vertices.  If  a  diagonal  and 
an  adjacent  side  are  chosen  as  axes,  what  are  the  coordinates  of  the 
vertices?  of  the  middle  points  of  the  sides?  of  the  center? 

8.  Find,  by  similar  triangles,  the  coordinates  of  the  point  which 
bisects  the  line  joining  the  points  (2,  7)  and  (4,  4). 

9.  Show  that  the  distance  from  the  origin  to  the  point  (a,  h)  is 
Va^  +  62.  How  far  from  the  origin  is  the  point  (a,  ~h)  ?  {-a,  b)  ? 
(-a,  ~b)l    (cf.  Art.  22.) 

10.  Prove,  by  similar  triangles,  that  the  points :   (2,  3),   (1,  ~3),  and 
(3,  9)  lie  on  the  same  straight  line. 

11.  Solve  exercises  1  to  4  and  10  if  the  coordinate  axes  make  an  angle 
of  60°.     Also  if  this  angle  be  45^ 

*  These  minus  signs  are  written  high  merely  to  indicate  that  they  are 
signs  of  quality  and  not  of  operation. 


22-23.1 


GEOMETRIC  CONCEPTIONS 


29 


23.  Polar  coordinates.  If  a  fixed  point  0  is  given  in  a 
fixed  directed  straight  line  OR,  then  the  position  of  any 
point  P  of  the  plane  will  be  fully  determined  by  its  distance 


Fig.  8.— 


^R 


OP  ==  p  from  the  fixed  point,  and  by  the  angle  0  which  the 
line  OP  makes  with  the  fixed  line. 

The  fixed  line  OR  is  called  the  initial  line  or  polar  axis,  the 
fixed  point  0  the  pole  of  the  system,  and  the  polar  coordinates 
of  the  point  P  are  the  radius  vector  p  and  the  directional  or 
vectorial  angle  0.  The  usual  rule  of  signs  applies  to  the 
vectorial  angle  6,  and  the  radius  vector  is  positive  if  meas- 
ured from  0  along  the  terminal  side  of  the  angle  6.  The 
point  P  is  designated  by  the  symbol  (p,  6). 

From  what  has  just  been  said  it  is  clear  that  one  pair  of 
polar  coordinates  (i.e.,  one  value  of  p  and  one  of  ^)  serve  to 
determine  one,  and  but  one,  point  of  the  plane.  On  the 
other  hand,  if  0  is  restricted  to  values  lying  between  0  and 
2  TT,  then  any  given  point  may  be  designated  hjfour  different 
pairs  of  coordinates.  P 


/-3 


240''   / 


i? 


R 


Fig.  9.-^ 


Fig.  9.-^  >^ 

^.^.,  the  polar  coordinates  (3,  60°)  determine  the  position 
of  the  point  P,  for  which  OP  =  3,  and  makes  an  angle  of  60° 


30 


ANALYTIC  GEOMETRY 


[Ch.  II. 


with  the  initial  line  0R\  but  the  same  point  may  be  given 
equally  well  by  the  pairs  of  coordinates :  (~3,  240°), 
(3,  -300°),  and  ("3,-120°);  and  so  in  general. 


-300>- 


FiQ.9.- 


/-3 


'120^ 


Fig.  9.^ 


R 


EXERCISES 

1.  Plot  accurately  the  following  points :  (2,  20°),   [2,  -\     (     7,  ^V 

(477,^),    (2,14  0,    (-1,-180°),    (7,-45°),    (-7,135°),    (s,  ?^), 

(^'f)'  (^'t)'  ^^'^°^'  ^°^  ^"^'^'^• 

2.  Construct    the   triangles   whose   vertices    are:   (2,-),    (3,—) 
/      5^\  \      8/      \       4  /> 
(1,  -7-)'^  fi^^^  by  measurement  the  lengths  of  the  sides  and  the  coordi- 
nates of  their  middle  points. 

3.  The  base  of  an  equilateral  triangle,  whose  side  is  5  inches,  is  taken 
as  the  polar  axis,  with  the  vertex  as  pole ;  find  the  coordinates  of  the 
other  two  vertices. 

4.  Write  three  other  pairs  of  coordinates  for  each  of  the  points 
(2,  1);  (-3,75°);  (5,0°);   (0,60°). 

5.  Where  is  the  point  whose  radius  vector  is  7  ?  "whose  radius  vector 
is  —7?  whose  vectorial  angle  is  25°?  whose  vectorial  angle  is  O^*")  ?  whose 
vectorial  angle  is  —  180°? 

6.  Express  each  of  the  conditions  of  Ex.  6  by  means  of  an  equation. 

7.  What  is  the  direction  of  the  line  through  the  points  ( 3,  —  ]  and 

24.  Notation.  In  the  following  pages,  to  secure  uniformity 
and  in  accordance  with  Art.  6,  a  variable  point  will  be  desig- 


23-26.] 


GEOMETRIC  CONCEPTIONS 


31 


nated  by  P,  and  its  coordinates  by  (x^  y)  or  (p,  ^).  If 
several  variable  points  are  under  consideration  at  the  same 
time,  tliey  will  be  designated  by  P,  P',  P" ,  P'",  •••,  and 
their  coordinates  by  (a;,  ?/),  (x' ,  y')^  (x" ^  y'),  (x"\  y"'^->  •••, 
or  by  (/.,  ^),  (p^  0%  (p",  0^'),  (p"',  6'"),  ....  Fixed  points 
will  be  designated  by  P^,  Pg,  •••,  and  their  coordinates  by 

(^1^  3/i)'  (%  «/2)'  •••'  01'  ^y  (/'i'  ^i)'  (^2^  ^2)'  •••• 


II.    ELEMENTARY   APPLICATIONS 

25.  The  methods  of  representing  a  point  in  a  plane  that 
have  been  adopted  in  the  previous  articles  lead  at  once  to 
several  easy  applications,  such  as  finding  the  distance  be- 
tween two  points,  the  area  of  a  triangle,  etc.  The  form  of 
the  results  will  depend  upon  the  particular  system  of  coordi- 
nates chosen,  but  the  method  is  the  same  in  each  case. 
Here,  as  in  the  more  difficult  problems  that  arise  later,  to 
gain  the  full  advantage  of  the  analytic  method  the  student 
should  freely  use  geometric  constructions  to  guide  his  alge- 
braic work,  but  he  should,  at  the  same  time,  see  clearly  that 
the  method  is  essentially  algebraic. 

26.  Distance  between  two  points. 

(1)  Polar  coordinates.  Let  OB  be  the  initial  line,*  0  the 
pole,  and  let  P^  =  (/o^,  6^}  and  Pg  =  (/Og,  ^2)  ^®  ^^^  ^^^  given 


Fig.  10. 


Fig.  10.-^ 


*  The  demonstration  applies  to  each  figure. 


32 


ANALYTIC  GEOMETRY. 


[Ch.  II. 


fixed  points.  It  is  required  to  find  the  distance  P^Pg  —  ^ 
in  terms  of  the  given  constants  /a^,  p^,  0^,  and  6^.  In  the 
triangle  OF^P^^  (cf.  Art.  16) 

P^2^  Oi\'  +  OF^-2  .  OF^  ■  OP^ .  cos  P^OP^. 


hence 


^'  =  Pi  +  /^2^  -  2  f)i/>2  cos  ((92  -  6'i), 
ci  =  Vpi»  +  P3»  -  2piP2  cos  (9^  -  0j) 


[1] 


(2)    Cartesian   coordinates ;    axes    not    rectangular.       Let 
OX  and  O^T  be  the  coordinate  axes,  meeting  at  an  angle 

Ya 

7 


Yl 

LA 


B 


.Q 


"IP. 


Fig.  11. 


10      M^  M, 

Fig. 11 A 


XOY=  CO*  and  let  P^  =  (x^,  y^  and  P^  =  (x^,  y^  be  the  two 
given  points  ;  it  is  required  to  find  the  distance  P^P^  =  d 
in  terms  of  x^^  x^,  ?/j,  y^^  and  o). 

Construction  :  Extend  the  abscissa  L^P^  of  the  point  P^ 
to  meet  the  ordinate  M^P^  of  the  point  P^,  in  Q  ;  then  in 
the  triangle  P^QP^^  (cf.  Art  16) 

F^Pi  -  F\Q^  -^QP^-2-P,Q-  QP,^ .  cos  P,QP,.  Fig.  11«, 
PJ^  =  p;q'  +  F;q^  -2.P,Q.  P,Q  .  cos  P,QP^,  Fig.  ll^ 

AA'  =QP?  +  P^  -2'QP^.  P,Q  .  cos  Pi§P2,  Fig.  11^ ; 
which  gives,  for  each  figure, 

^=  V(Xi  -  i»2)3  +  (2/1  -  1/3)2+  2  (iCi  -  a?3)(i/i  -  t/3)C0S  (O.t 

*  The  demonstration  applies  to  each  figure. 

t  By  examining  other  possible  constructions  the  student  should  assure 
himself  of  the  generality  of  this  formula. 


26-27.]  GEOMETRIC  CONCEPTIONS  33 

(3)  Hectmigular  coordinates.  If  co  =  — ,  i,e.  if  the  coordi- 
nate axes  are  rectangular,  then  cos  o)  =  0,  and  the  formula 
for  the  distance  between  the  two  given  points  becomes 


<Z=V(a?i-a?2)-'^  +  (2/i-2/3)^  ...  [2] 
Since  either  of  the  two  points  may  be  named  Pj,  this  formula 
may  be  expressed  in  words  thus  :  In  rectangular  coordinates^ 
the  square  of  the  distance  between  two  given  points  is  the  square 
of  the  difference  between  their  abscissas  plus  the  square  of  the 
difference  betiveen  their  ordinates. 

27.  Slope  of  a  line.  By  the  slope  of  a  line  is  meant  the 
tangent  of  the  angle  which  the  line  makes  with  the  positive 
end  of  the  o^-axis.* 

From  this  definition  it  at  once  follows  that  the  slope  m  of 
the  line  joining  the  two  points  Pi  =  (a^i,  y{)  and  Pg  =  fe?  ^2)? 

QP 

the  axes  being  rectangular,  is_w  =  -p— ^ ;  that  is, 

2/2  -  2/1  ron 

m  = .  .  .  \6\ 

0C2  —  oci  ^  -■ 

EXERCISES 

1.  Find  the  distances  between  the  points  (1,  3),  (2,  7),  and  (~4,  -4), 
taken  in  pairs. 

2.  Find  the  distances  for  the  points  of  Ex.  1,  if  the  axes  are  oblique 
with  CO  =  60°. 

3.  Prove  that  the  points     (-2,-1),  (1,0),  (4,3),  and  (1,2)  are  the 
vertices  of  a  parallelogram. 

4.  Find    the     distance    between    the    points    (a  +  &,    c  +  a}    and 
(c  -\-  a,  b  +  c)  ]  also  between   (a,  b)  and  ("a,  ~b). 

5.  Find  the   distances  between  the   points    (2,  30°),   [3,  —  j,and 

1  57r\  , ,     .      .  V     4  y 

1,  —  1,  taken  m  pairs. 


*  The  slope  of  a  roof  or  of  a  hill  has  the  same  meanhig.  Thus  if  the 
slope  of  a  hill  (to  the  horizontal)  is  yj^,  it  rises  3  feet  vertical  in  100  feet 
horizontal. 


TAN.  AX.   GEOM. 


34  ANALYTIC  GEOMETRY  [Ch.  II. 

6.  Prove  that  the  points  (0,  0''),  f  3,  ^J,  and  (3,  -)  form  an  equi- 
lateral triangle. 

7.  One  end  of  a  line  whose  length  is  13  is  at  the  point  (~"4,  8),  the 
ordinate  of  the  other  end  is  3 ;  what  is  its  abscissa? 

8.  Express  by  an  equation  the  fact  that  the  point  P  =  {x,  y)  is  at  the 
distance  3  from  the  point  (-2,  3)  ;  from  the  point    (0,0). 

9.  Express   by  an  equation   the   fact   that  the  point  P=(x,  y)  is 
equidistant  from  the  points  ("2,  3)  and  (7,  5). 

10.  Find  the  slopes  of  the  lines  which  join  the  following  pairs  of 
points  :  (3,  8)  and  ("1,  4)  ;  (2,  "3)  and  (7,  9)  ;  (1,  -4)  and  ("3,  5)  ;  (4,  -2) 
and  (-2,  -1). 

28.  One  great  advantage  of  the  analytic  method  of  solv- 
ing problems  lies  in  the  fact  that  the  analytic  results  which 
are  obtained  from  the  simplest  arrangement  of  the  geometric 
figure  with  reference  to  the  coordinate  axes  are,  from  the 
very  nature  of  the  method,  equally  true  for  all  other  arrange- 
ments. Thus  formulas  [1],  [2],  and  [3]  can  be  most  readily- 
obtained  if  the  points  are  all  taken  in  quadrant  I,  i.e.^  with 
their  coordinates  all  positive  ;  but  because  of  the  convention 
adopted  concerning  the  signs  as  essential  parts  of  the  coordi- 
nates, these  formulas  remain  true  for  all  possible  positions  of 
Pi  and  P2.  By  drawing  the  figures  and  making  the  proofs 
when  Pi  and  P2  are  taken  in  various  other  positions,  the 
student  should  assure  himself  of  the  generality  of  formulas 
[1],  [2],  and  [3]  of  articles  26  and  27. 

29.  The  area  of  a  triangle. 

1.  Rectangular  coordinates.  Given  a  triangle  with  the 
vertices  Pi  =  (2:1,  y{),  P^  =  fe,  ^2).  and  P3  =  (x^,  ^/g);  to  find 
its  area  in  terms  of  a^i,  X2,  x^^  ?/i,  1/2-,  and  y.^.  Draw  the  ordi- 
nates  itfiPi,  M2P2,  and  M^P^^  —  in  the  second  figure  extend 
iHfjPi  and  M2P2  to  meet  a  line  through  P2  parallel  to  the 
a:-axis.  If  A  represents  the  area  of  the  triangle  in  the  first 
figure,  then : 


27-29.] 


GEOMETRIC  CONCEPTIONS 


35 


Y^ 

k 

R 

\, 

p/- 

0 

' 

X. 

M, 

M, 

M.        ' 

Fig.  ^A 

Fig.  13.— 

A  =  P,M,M,P,  +  P,M,M,P,  -  P,M,M,P,, 
but     P,M,M,P,  =  l(^M,P,  +  M,P,) .  lfi.lf3  =  Kyi  +  ^3)(^'3-^i), 

and    P,M^M,P,=  l(iM,P,^-M,P,^  .  ilt/ii^f2=K^i  +  ^2)(^2-^i). 

•••      A  =  lKyi  +  ^3)(^3-^l)  +  (y2  +  ^3)(^2-^3) 

-(^1+^2)  (^2-^0! 

=  JK^l  +  ^2)(^1  -  ^2)  +  (^2  +  ^3)  (^2  -  ^3) 

+  (y3  +  yi)(^3-^l)S      • 

This  may  also  be  written  in  the  form 

A  =  ^  {^1(^2  -  ^3) +^2(^3  -  yi)  +  ^3(^1  -y^]*' 

So  also  if  Ai  represents  the  area  of  the  triangle  in  the 
second  figure,  then 

Ai=  P,E,H,P,  -  P,H,P,  -  P,H,P, 

=  1  K^i^i  +  ^3i"3)  •  M,M,-H,P,  .  H,P,-H,P, .  P,H,l 

=  i  K^i  -  !^2 + ^3  -  ^2)  (^3  -  ^1)  -  iyi  -  ^2)  (^2  -  ^1) 

-  (^3 - ^2)  (^3 -^2)U   [^1^  ^25  and  ;«/2  being  negative] 
=  i  1^1(^2-^3) +2:2(^3-^1) +^3(^1-^2)!,*  as  above  [4a]. 


*  In  the  determinant  notation  this  may  be  written 


[4  a] 


area  of  the  triangle 


36 


AJ^^ALYTIC  GEOMETRY 


[Ch.  II. 


If,  instead  of  rectangular  coordinate  axes,  oblique  axes 
making  an  angle  XOY=(o  had  been  used,  it  would  have 
been  necessary  merely  to  multiply  the  second  members  in 
the  results  just  found  by  sin  ©  in  order  to  express  the  areas 
of  the  triangles. 

2.  Polar  coordinates. 
Let  the  vertices  of  the 
triangle  be  Pi  =  (p^,  (9i), 
P.  =  (p2,  O2),  and  Pg  ~ 
Cps^  ^3)?  to  find  its  area 
A  in  terms  of  pi,  p2,  ps^  ^1, 
^21  aiid  02' 


Fig.  13. 


Manifestly,     A  =  OP.P,+  OP^P,-  OP.P^, 

but     OP.2P2  =  ^P2Pz  sin((93-^2),  OPzPi  =  Ip^o^  sin(6>i-^3), 

and  0P2Pi  =  l  P2P1  sin  (Oi  —  O^)- 

•'•   ^=  J  J/32P3sin(^3-6'2)+/93/3isin(6>i-(93)-/32Pisin((9i-6'2)i, 
which  may  also  be  written 

A  =  i  \P1P2  sin  (^2-^1)  +  ^2^3  sin  (6^3  -  ^2) 

+/03P1  sin((9i-6'3)S.     .     .     .     [5] 

The  symmetry*  in  formulas  [4],  [4a],  and  [5]  should  be 
carefully  noted ;  it  may  be  remarked  also,  that  in  the  appli- 
cation of  these  formulas  to  numerical  examples,  the  resulting 
areas  will  be  positive  or  negative  according  to  the  relative 
order  in  which  the  vertices  are  named. 

*  This  kind  of  symmetry  is  known  as  cyclic  (or  circular)  symmetry.     If 

1/^3 
the  numbers  1,2,  and  3  be  arranged  thus  \kJj-  ,  then  the  subscripts  in  the 

•7 

first  term  (in  [4a]  say)  begin  with  1  and  follow  the  arrow  heads  around  the 

circle  {i.e.  their  order  is  1,  2,  3),  those  of  the  second  term  begin  with  2  and 

follow  the  arrow  heads  (their  order  is  2,  3,  1),  and  those  of  the  third  term 

begin  with  3  and  follow  the  arrow  heads. 


29-30.] 


GEOMETRIC  CONCEPTIONS 


37 


EXERCISES 

1.  Find  the  areas  of  the  following  triangles:  (1)  vertices  at  the 
points  (3,  5),  (4,  2),  and  (1,  3);  (2)  vertices  at  the  points  (7,  3),  (4,  6), 
and  (3,  -2)  ;    (3)  vertices  at  the  points  (11,  9),  (0,  "2),  and  (-5,  3). 

Solve  without  using  the  formula,  and  then  verify  by  substituting  in 
the  formula. 

2.  Prove  that  the  area  of  the  triangle  whose  vertices  are  at  the  points 
(2,  3),  (5,  4),  and  (-4,  1)  is  zero,, and  hence  that  these  points  all  lie  on 
the  same  straight  line. 

3.  Do  the  points  (2,  3),  (1,  "3),  and  (3,  9)  lie  on  one  straight  line? 
(cf.  Ex.  10,  p.  28.) 

4.  Do  the  points  (7,  30°),  (0,  0°),  and  (-11,  210°)  lie  on  one  straight 
line?  Solve  this  by  showing  that  the  area  of  the  triangle  is  zero,  and 
then  verify  by  plotting  the  figure. 

5.  Find  the  area  of  the  triangle  (  tt,  -     j ,  (  2  tt,  -     j ,  and  (  —  tt,  ^     J . 

6.  Derive  formula  [4]  when  P^  is  in  quadrant  II,  P^  in  quadrant  III, 
and  P3  in  quadrant  IV. 

7.  Find  the  area  of  the  first  two  triangles  in  Ex.  1  if  the  axes  make 
an  angle  of  60°  with  each  other. 


30.  To  find  the  coordinates  of  the  point  which  divides  in 
a  given  ratio  the  straight  line  from  one  given  point  to 
another.  Let  P^  =  (x-^^  y-^  and  P^  =  (x^,  y^  be  the  two 
given  points,  P<^^{x^^  y^  the  required  point,  and  let  the 


FiG.U.- 


FiG.  u.- 


ratio  of  the  parts  into  which  Pg  divides  P^P^  he  m-^  :  mcy^ ; 
^.e.,  let  PjPg  :  P3P2  —  ^^1  *  '^h'     I^^^w  the  ordinates  M^P^^ 


38  ANALYTIC  GEOMETRY  [Ch.  II, 

Mc^P^,  M^P^,  and,  through  P^  and  Pg,  draw  lines  parallel  to 
OX,  meeting  M^P^  and  M^P^  in  R  and  Q  respectively. 

To  find  OM^  =  Xq  and  M^P^  =  t/^  in  terms  of  x^,  x^,  y^,  y^, 
TTij,  and  m^. 

The  triangles  P^RP^  and  P^QP^  are  similar; 

P,R  _  RP,  _  P,P, 


therefore 


But 


P^Q      QP.^     P^P^ 


P,P 


m. 


P^P^        ^«2 


X. 


3' 


and  P^R  =  ^3  —  ^p  P%Q  =  ^2  ~~ 

I^Pz  =  3/3  -  Vv  QPi  =  ^2  -  ^3- 
[In  Fig.  14  (^)),  a;p  i/^,  y^,  and  2/3  are  negative.]     . 

theretore  — — —  —      _      —  — - ' 

*^2    *^3    y^    y^      2 

whence 

a?3  = ^ ^ —  and  2/3  = ^ .  .      .       o 

.  The  above  reasoning  applies  equally  well  whatever  the 
value  of  ft)  (the  angle  made  by  the  coordinate  axes),  hence 
formulas  [6]  hold  whether  the  axes  be  rectangular  or  oblique. 
Formulas  [6]  were  obtained  on  the  implied  hypothesis 
that  P3  lies  between  P^  and  P^  ;  i.e.,  that  Pg  is  an  inte7mal 
point  of  division.  If  P3  is  taken  in  the  line  P1P2  produced, 
and  not  between  P^  and  P^,  it  still  forms,  with  P^  and  P^, 
two  segments  P1P3  and  P^P^^  and  P3  may  be  so  taken  that, 
numerically,  the  ratio  of  P1P3  :  P?,^^  ^^^^J  bave  any  real 
value  whatever ;  but  the  sign  of  this  ratio  is  negative  when 
Pg  is  not  between  P^  and  Pg,  for,  in  that  case,  the  segments 
PjPg  and  P3P2  have  opposite  directions.  Hence,  to  find 
the  coordinates  of  that  point  which  divides  a  line  externally 
into  segments  whose  numerical  ratio  is  m-^  :  m,^,  it  is  only 


30.]  GEOMETRIC  CONCEPTIONS  39 

necessary  to  prefix  the  minus  sign  to  either  one  of  the  two 
numbers  m^  or  m^  in  formulas  [6].  These  formulas  then 
become 

^  m^x^  -  m^x^         ^rn^y^.  -^2^1.      .     .      [-71 


Cor.     If  -P3  be  the  middle  point  of  P^P^^  then  m-^  = 
and  formulas  [6]  become 


w 


2' 


a^s  -  —  2~'  ^3  -  ~^~  ;       .       .       .       [8] 

^.g.,  the  abscissa  of  the  middle  point  of  the  line  joining  two 
given  points  is  half  the  sum  of  the  abscissas  of  those  points, 
and  the  ordinate  is  half  the  sum  of  their  ordinates. 

The  remarks  in  Art.  28  are  well  illustrated  by  formulas 
[4]  to  [8]; 

EXERCISES 

1.  By  means  of  an  appropriate  figure,  derive  formulas  [7]  independ- 
ently of  [6]. 

2.  The  point  Pg  =  (2,  3)  is  one  third  of  the  distance  from  the  point 
P^={-1,  4)  to  the  point  Po={x2,  2/2)  ;    to  find  the  coordinates  of  Pg- 

Here  P^  and  P3  are  given,  with  a;^  =  —  1,  ?/i  =  4,  x^  =  2,  y^  =  3,  also 
Tn■^^  =  1,  and  7/12  =  2 ;   therefore,  from  [6], 

g_^2  +  2(-l)       ^^^  3_y.  +  2(4) 

which  give  x^  =  S  and  ^2  =  1 1  therefore  the  required  point  Pg  is  (8,  1). 

3.  Find  the  points  of  trisection  of  the  line  joining  (1,  -2)  to  (3,  4). 

4.  Find  the  point  which  divides  the  line  from  (1,  3)  to  (-2,  4) 
externally  into  segments  whose  numerical  ratio  is  3  :  4. 

Here  x^  =  1,  y^  =  3,  Xg  =  —  2,  ^3  =  4,  m^  =  3,  and  m^  =  4,  but  the 
point  of  division  being  an  external  one,  the  two  segments  are  oppo- 
sitely directed ;  therefore  one  of  the  numbers  3  or  4,  say  4,  must  have 
the  minus  sign  prefixed  to  it.     Substituting  these  values  in  [6], 

3(l)-4(-2)^_        ^^^  3(3)-4(4)^ 

3  3  _  ^  ys  3  _  ^  «  , 

the  required  division  point  is,  therefore,  Pg  =  (~11,  7). 


40  ANALYTIC  GEOMETRY  [Ch.  II. 

The  same  result  would  have  been  obtained  had  m.^  —  3,  instead  of 
TWg  =  4,  been  given  the  minus  sign  ;  or,  again,  formulas  [7]  could  have 
been  employed  to  solve  this  jDi'oblem. 

5.  Solve  Ex.  4  directly  from  a  figure,  without  using  either  [6]  or  [7]. 

6.  Find  the  points  which  divide  the  line  from  (1,  5)  to  (2,  7)  inter- 
nally and  externally  into  segments  which  are  in  the  ratio  2  : 3. 

7.  A  line  AB  is  produced  to  C,  so  that  BC  =  ^AB;  if  the  points  A 
and  B  have  the  coordinates  (5,  6)  and  (7,  2),  respectively,  what  are  the 
coordinates  of  C  ? 

8.  Prove,  by  means  of  Art.  30,  that  the  median  lines  of  a  triangle 
meet  in  a  point,  which  is  for  each  median  the  point  of  trisection  nearest 
the  side  of  the  triangle. 

31.   Fundamental    problems   of   analytic   geometry.       The 

elementary  applications  already  considered  have  indicated 
how  algebra  may  be  applied  to  the  solution  of  geometric 
problems.  Points  in  a  plane  have  been  identified  with  pairs 
of  numbers,  —  the  coordinates  of  those  points,  —  and  it  has 
been  seen  that  definite  relations  between  such  points  corre- 
spond to  definite  Telations  between  their  coordinates. 

It  will  be  found  also  that  the  relation  between  points, 
which  consists  in  their  lying  on  a  definite  curve,  corre- 
sponds to  the  relation  between  their  coordinates,  which 
consists  in  their  satisfying  a  definite  equation.  From  this 
fact  arise  the  two  fundamental  problems  of  analytic  geom- 
etry : 

I.  Given  an  equation^  to  find  the  corresponding  geometric 
curve,  or  locus. 

II.  Griven  a  geometric  curve,  to  find  the  corresponding 
equation. 

When  this  relation  between  a  curve  and  its  equation  has 
been  studied,  then  a  third  problem  arises  : 

III.  To  find  the  properties  of  the  curve  from  those  of  its 
equation.  ,. 


30-31.]  GEOMETRIC  CONCEPTIONS  41 

The  first  two  problems  will  be  treated  in  the  two  succeed- 
ing chapters,  while  the  remaining  chapters  of  Part  I  will 
be  concerned  chiefly  with  the  third  problem.  In  this  appli- 
cation of  analytic  methods,  however,  only  algebraic  equa- 
tions of  the  first  and  second  degrees  will  for  the  most  part 
be  considered.  In  Chapter  XIII  is  given  a  brief  study  of 
other  important  equations  and  curves. 

EXAMPLES  ON   CHAPTER   II 

1.  Find  the  area  of  the  quadrilateral  whose  vertices  are  the  points 
(1,  0),  (3,  I),  (-1,  16),  and  (-4,  2).     Draw  the  figure. 

2.  Find  the  lengths  of  the  sides  and  the  altitude  of  the  isosceles 
triangle  (1,  5),  (5,  1),  (-9,  ~9).  Find  the  area  by  two  different  methods, 
so  that  the  results  will  each  be  a  check  on  the  other. 

3.  Find  the  coordinates  of  the  point  that  divides  the  line  from  (2,  3) 
to  (~1,  ~6)  in  the  ratio  3:4;  in  the  ratio  2  : -3 ;  in  the  ratio  3  :  ~2. 
Draw  each  figure. 

4.  One  extremity  of  a  straight  line  is  at  the  point  (~3,  4),  and  the 
line  is  divided  by  the  point  (1,  6)  in  the  ratio  2:3;  find  the  other  ex- 
tremity of  the  line. 

5.  The  line  from  (-6,  ~2)  to  (3,  —1)  is  divided  in  the  ratio  4:5;  find 
the  distance  of  the  point  of  division  from  the  point  (~4,  6). 

6.  Find  the  area  and  also  the  perimeter  of  the  triangle  whose  vertices 
are  the  points  (3,  60°),  (5,  120°),  and  (8,  30°). 

7.  Show  analytically  that  the  figure  formed  by  joining  the  middle 
points  of  the  sides  of  any  quadrilateral  is  a  parallelogram. 

8.  Show  that  the  points  (1,  3),  (2,  V6),  and  (2,  —VQ)  are  equidis- 
tant from  the  origin. 

9.  Show  that  the  points  (1,  1),  (— 1,  —1),  and  ("Vs,  -V3)  form  an 
equilateral  triangle.     Find  the  slopes  of  its  sides. 

10.  Prove  analytically  that  the  diagonals  of  a  rectangle  are  equal. 

11.  Show  that  the  points  (0,  -1),  (2,  1),  (0,  3),  and  (-2,  1)  are  the 
vertices  of  a  square. 


42  ANALYTIC  GEOMETRY  [Ch.  II.  31. 

12.  Express  by  an  equation  that  the  point  (h,  k)  is  equidistant  from 
(-1,  1)  and  (1,  2);  from  (1,  2)  and  (1,  -2).  Then  show  that  the  point 
(I,  0)  is  equidistant  from  (-1,  1),  (1,  2),  and  (1,  -2). 

13.  Prove  analytically  that  the  middle  point  of  the  hypotenuse  of 
a  right  triangle  is  equidistant  from  the  three  vertices. 

14.  Three  vertices  of  a  parallelogram  are  (1,2),  (-5,-3),  and  (7,  -6)  ; 
what  is  the  fourth  vertex  ? 

15.  The  center  of  gravity  of  a  triangle  is  at  the  point  in  which  the 
medians  intersect.  Find  the  center  of  gravity  of  the  triangle  whose 
vertices  are  (2,  3),  (4,  -5),  and  (3,  -6).     (cf.  Ex.  8,  p.  40.) 

16.  The  line  from  (x^  y^  to  {x^,  y^  is  divided  into  five  equal  parts ; 
find  the  points  of  division. 

17.  Prove  analytically  that  the  two  straight  lines  which  join  the 
middle  points  of  the  opposite  sides  of  a  quadrilateral  mutually  bisect 
each  other. 

18.  Prove  that  (1,  5)  is  on  the  line  joining  the  points  (0,  2)  and  (2,  8), 
and  is  equidistant  from  them. 

19.  If  the  angle  between  the  axes  is  30°,  find  the  perimeter  of  the 
triangle  whose  vertices  are  (2,  2),  (-7,  -1),  and  (-1,  1).     Plot  the  figure. 

20.  Show  analytically  that  the  line  joining  the  middle  points  of  two 
sides  of  a  triangle  is  half  the  length  of  the  third  side. 

21.  A  point  is  7  units  distant  from  the  origin  and  is  equidistant  from 
the  points  (2,  1)  and  (-2,  -1)  ;  find  its  coordinates. 

22.  Prove  that  the  points  (a,  6  +  c),  {h,  c  +  a),  and  (c,  «-}-&)  lie  on 
the  same  straight  line.     (cf.  Ex.  2,  p.  37.) 


CHAPTER   III 
THE  LOCUS  OF  AN  EQUATION 

32.  The  locus  of  an  equation.  A  pair  of  numbers  x^  y  is 
represented  geometrically  by  a  point  in  a  plane.  If  these 
two  numbers  (x^  y)  are  variables,  but  connected  by  an  equa- 
tion, then  this  equation  can,  in  general,  be  satisfied  by  an 
infinite  number  of  pairs  of  values  of  x  and  ?/,  and  each  pair 
may  be  represented  by  a  point.  These  points  will  not, 
however,  be  scattered  indiscriminately  over  the  plane,  but 
will  all  lie  in  a  definite  curve,  whose  form  depends  only 
upon  the  nature  of  the  equation  under  consideration ;  and 
this  curve  will  contain  no  points  except  those  whose  co- 
ordinates are  pairs  of  values  which  when  substituted  for 
X  and  ?/,  satisfy  the  given  equation.  This  curve  is  called 
the  locus  or  graph  of  the  equation ;  and  the  first  funda- 
mental problem  of  analytic  geometry  is  to  find,  for  a  given 
equation,  its  graph  or  locus. 

33.  Illustrative  examples  :  Cartesian  coordinates. 

(1)    Given  the  equation  x  -{•  o  =  0,  to  find  its  locus.     This  equation  is 

satisfied    by    the    pairs    of    values    x^=  —  o,    y^  =  2;    a^g  =  —  5,    3/2  =  8; 

arg  =  —  5,  2/3  =  —  2  ;  etc.,  that  is,  by  every  pair  of  values  for  which  x  =  —  o. 

Such  points  as 

P,=  (x„y,)  =  i-D,2), 

Po=(x,,,  2/2)  =  (-5,  3), 

^3  =  (^3'  3/3)  =  (r^^y  ~3)»  etc., 
all  lie  on  the  line  MN,  parallel  to  the  y-axis,  and  at  the   distance  5  on 
the  negative  side  of  it,  —  this  line  extending  indefinitely  in  both  direc- 

43 


44 


ANALYTIC  GEOMETRY 


[Ch.  III. 


tions.      Moreover,  each   point   of   MN  has   for   its   abscissa  -o,  hence 
the  coordinates  of   each  of   its  points   satisfy  the   equation   x  +  5  ==  0. 

In  the  chosen  system  of  coordi- 
N 


N 


C. 


i? 


A    Pa 


M 


O 


D 


B 


Fig. 15. 


M 


nates,  the  line  MN  is  called  the 
locus  of  this  equation. 

Similarly,  the  equation  a;  — 5 
=  0  is  satisfied  by  any  pair  of 
values  of  which  x  is  5,  such  as 
(5,  2),  (5,  3),  (5,  4),  etc. ;  all  the 
corresponding  points  lie  on  a 
straight  line  M'N',  parallel  to 
the  ?/-axis,  at  the  distance  5  from 
it,  and  on  its  positive  side ;  i.e., 
M'N'  is  the  locus  of  the  equa- 
tion X  —  5  =  0. 

(2)  Given  the  equations  y  ±  3  =  0,  to  find  their  loci  By  the  same 
reasoning  as  in  (1)  it  may  be  shown  that  the  locus  of  the  equation 
?/  -f  3  =  0  is  the  straight  line  AB,  parallel  to  the  x-axis,  situated  at  the 
distance  3  from  it,  and  on  its  negative  side.  Also  that  the  locus  of  the 
equation  ?/  —  3  =  0  is  CD,  a  line  parallel  to  the  a:-axis,  at  the  distance 
3  from  it,  and  on  its  positive  side. 

More  generally,  it  is  evident  that  in  Cartesian  coordinates  (rectangular 
or  oUique),  an  equation  of  the  first  degree,  and  containing  hut  one  variable^ 
represents  a  straight  line  parallel  to  one  of  the  coordinate  axes. 

(3)  Given  the  equation  3  a;  —  2  ?/  +  12  =  0,  to  find  its  locus.  In  this 
equation  both  the  variables  appear.  By  assigning  any  definite  value  to 
either  one  of  the  variables,  and  solving  the  equation  for  the  other,  a  pair 
of  values  that  will  satisfy  the  equation  is  ob- 
tained. Thus  the  following  pairs  of  values 
are  found : 


^1  =  0,  ?/i  =  6 

a^2  ^^  J-  ?  y 2  ^^     2 

arg  =  2,  ?/3  =  9 

3^4  =  ^'  2/4  =  10^ 


+   CO,     ?/   =+   CO 


^5  =  -  1»  y.5  =  4^ 

Xg  =  -  2,  ?/^,  =  3 

^r  =  -  ^>  2/r  =  H 

2^8  =  -  '^'  ^8=0 

a:  =  —  CO,  V  = 


y  =-  CO 

Plotting  the  corresponding  points 

Pj,  P„  P„  P, ... ,  where  P^  =  {x„  y,)  =  (0,  6), 

P.->  =  ix^,y^^  =  {\,  7i),  etc., 
they  are  all  found  to  lie  on  the  straight  line  EF,  which  is  the  locus  of 
the  equation  3  a;  —  2  ?/  -f  12  =  0. 


Fia.16. 


33.] 


THE  LOCUS   OF  AN  EQUATION 


45 


In  Chap.  V,  it  will  be  shown  that,  in  Cartesian  coordinates,  an  equar 
tion  of  the  first  degree  in  two  variables  always  represents  a  straight  line. 

(4)  Given  the  equation  ?/2  =  4  x,  to  find  its  locus.  This  equation  is 
satisfied  by  each  of  the  following  pairs  of  values,  found  as  in  (3)  above : 

^'i  =  0,  ?/i  =  0 

^2  =  1'   2/2  =  +  2 

^3  =  1'  2/3  =  -  2_ 

x^  =  2,  3/4  =  2  ^/2  =  2.8,  approximately 
Xg  =  2,  2/5  =  —  2  V2=  —2.8,  approximately 
^6  =  ^5  2/6  =  +  4 


x  =  +  oo,  y  =  ±.  (X) 

and  for  any  negative  value  of  x  the  corre- 
sponding value  of  y  is  imaginary. 

The  corresponding  points  are  :  ' 

Pi  =  (0,  0),  P2  =  (1.2),  P3  =  (l,-2),  etc.  FiG.17.     P/ 

All  these  points  are  found  to  lie  on  the  curve  as  plotted  in  Fig.  17. 
This  curve  is  called  a  paraLola,  and  will  be  studied  in  a  later  chapter. 

The  parabola  is  one  of  the  curves  obtained  by  the  intersection  of  a 
circular  cone  and  a  plane,  (cf.  Appendix,  Note  D.)  It  will  be  shown 
in  Chap.  XII  that  in  Cartesian  coordinates,  the  locus  of  any  alge- 
braic equation  in  two  variables  and 
of  the  second  degree  is  a  "conic  sec- 
tion." 

(5)  Given  the  equation,  y  =  25  log  x, 
to  find  its  locus.  A  table  of  logarithms 
shows  that  this  equation  is  satisfied  by 
the  following  pairs  of  values  : 


x^  =  0, 

2/1=-^ 

xj  =    6, 

2/7   =  19-4 

X2  =  1) 

^2  =  0 

xs   =    7, 

2/8   =21.1 

Xg  =  2, 

2/3  =  7.5 

x^  =  10, 

2/9   =25 

x^  =  3, 

2/4  =  11-9 

-^10  =  lOj 

2/10  =  29.4 

x,  =  4, 

2/5  =  15 

X,,  =  20, 

2/11  =  32.5 

Xn    =     5, 

y.  =  17.5 

etc. 

etc. 

The  corresponding  points  are  : 
P,  =  (0,  -  CO),  P,  =  (h  0),  P3  =  (2,  7.5), 
etc. ;  and  the  locus  of  the  above  equa- 
tion   is    approximately   given    by    the 


curve  drawn  through  these  points  as  shown  in  Fig.  18. 


46 


ANALYTIC  GEOMETRY 


[Ch.  III. 


(6)  Given  the  equation  y  =  tan  x,  to  Jind  its  locus.  By  means  of  a  table 
of  "  natural "  tangents  it  is  seen  that  this  equation  is  satisfied  by  the 
following  pairs  of  values  of  x  and  y : 


Dkgrees 

Eadians 

x^  =    0 

=  0.00 

yi  =0 

X,    =10 

=  0.17 

y,  =  0.18 

X,  =20 

=  0.35 

y,  =0.36 

x^  =30 

=  0.52 

y^  =  0.58 

^5  =  40 

=  0.70 

y,  =  0.84 

X,  =50 

=  0.87 

3/6  -  1-19 

Xj  =60 

=  1.05 

yr  =  1-73 

Xg  =  70 

=  1.22 

yg  =2.75 

Xg  =80 

=  1.40 

2^9  =5.67 

X,,  =  90 

=  1.57 

2/io  =  ^ 

X,,  =  -  10 

=  -  0.17 

^11  =  -  0-18 

x^^  =  -  20 

=  -  0.35 

3/i2  =  -  0-36 

a:j3  =  -  30 

=  -  0.52 

3/13  =  -  0.58 

etc. 

etc. 

etc. 

The  corresponding  points  are  : 

Pi  =  (0,  0),  P^  =  (O.U,  0.18),  P3  =  (0.3.5,  0.36),  etc., 
and  the  locus  is  approximately  as  shown  in  Fig.  19. 


Fig. 19. 


X 


34.  Loci  by  polar  coordinates.  Analogous  results  are  obtained  for  a 
System  of  polar  coordinates,  as  will  be  best  seen  from  an  example. 
Given  the  equation  p  =  4:Cos  0,  to  Jind  its  locus. 


33-35.]  THE  LOCUS   OF  AN  EQUATION  47 

This  equation  is  satisfied  by  the  following  pairs  of  values,  found  as  in 
Art.  33  (3)  and  (4)  :  ' 

0,  =  S0°  p2  =  2^3  =  3.46  + 

^,  =  60^  p3  =  2 

0^  =  4:5°  p4  =  2V2  =  2.8  + 

^5  =  90°  p,  =  0 

6*6  =  -30°  p6  =  3.46  + 

^^  =  -  60°  p^  =  2 

^3  =---45°  p,  =  2.8-{-     ■ 

0,  =  -90°  p,  =  0 

etc.  etc. 

The  corresponding  points  are  : 

Pi=(4,  0°);  P2=(3.46  +  ,  30°);  P,  =  (2,  60°);  P,=  C2.S  +  ,  45°); 
P5=P9=  the  pole  O=(0,  ±90°);  Pg=(3.46  +  ,  -30");  Pj  =  (2,  -60°); 
etc. 

All  these  points  are  found  to  lie  on  the  circumference  of  a  circle 
whose  radius  is  2,  the  pole  being  on  the  circumference,  and  the  polar  axis 
being  a  diameter.     This  circle  is  the  locus  of  the  equation  p  =  4  cos  0. 

EXERCISES 

Plot  the  loci  of  the  following  equations : 


1. 

x  =  0. 

7. 

x2  +  y^  =  4. 

13.    ^2  +  ^2  3^  9, 

2. 

y  =  0. 

8. 

a;  +  y  =  4. 

14.  u'^+v=0. 

3. 

hx  =  Q. 

9. 

X  —  y  =  0. 

15.  s  =  16  t\ 

4. 

3a;  =  7. 

10. 

r^1_y1  =  4. 

16.^  +  ^  =  1. 
2      3 

5. 

2y  +  b 

=  0. 

11. 

2  a:2  +  2/2  =  4. 

17.  p  =  3. 

6. 

x  +  y  = 

0. 

12. 

V  =  32 1. 

18.  p  cos  ((9 -40°)  = 

19.  y  =  -  xK 

5. 

35.  The  locus  of  an  equation.  By  the  process  illustrated 
above,  of  constructing  a  curve  from  its  equation,  the  first 
conception  of  a  locus  is  obtained,  viz. : 

(1)  The  locus  of  an  equation  co7itainin(j  two  variables  is 
the  line^  or  set  of  lines^  which  coiitains  all  the  points  tvhose 
coordinates  satisfy/  the  given  equatio7i^  aiid  ivhich  contaijis 
no   other  points.     It  is  the  place  ivhere  all  the  points,  and 


48  ANALYTIC  GEOMETRY  [Ch.  III. 

only  those  points,  are  found  whose  coordinates  satisfy  the 
given  equation. 

A  second  conception  of  the  locus  of  an  equation  comes 
directly  from  this  one,  for  the  line  or  set  of  lines  may  be 
regarded  as  the  path  traced  by  a  point  which  moves  along 
it.  The  path  of  the  moving  point  is  determined  by  the 
condition  that  its  coordinates  for  every  position  through 
which  it  passes  must  satisfy  the  given  equation.  Thus  the 
line  EF  (the  locus  of  eq.  (3),  Art.  33)  may  be  regarded 
as  the  path  traced  by  the  point  P,  which  moves  so  that 
its  coordinates  (a;,  ?/)  always  satisfy  the  equation 

3a;-2^  +  12  =  0. 

Thus  arises  a  second  conception  of  a  locus,  viz. : 

(2)  The  locus  of  an  equation  is  the  path  traced  hy  a  point 
which  moves  so  that  its  coordinates  always  satisfy  the  given 
equation. 

In  either  conception  of  a  locus,  the  essential  condition 
that  a  point  shall  lie  on  the  locus  of  a  given  equation  is, 
that  the  coordinates  of  the  point  ivhen  substituted  respectively 
for  the  variables  of  the  equation^  shall  satisfy  the  equation; 
and  in  order  that  a  curve  may  be  the  locus  of  an  equa- 
tion, it  is  necessary  that  there  be  no  other  points  than  those 
of  this  curve  ivhose  coordinates  satisfy  the  equation. 

36.  Classification  of  loci.  The  form  of  a  locus  depends 
upon  the  nature  of  its  equation ;  the  curve  may  therefore 
be  classified  according  to  its  equation,  an  algebraic  curve 
being  one  whose  equation  is  algebraic,  and  a  transcendental 
curve  one  whose  equation  is  transcendental.  In  particular, 
the  degree  of  an  algebraic  curve  is  defined  to  be  the  same 
as   the   degree   of   its   equation.      The  following  pages  are 


30-37.J  THE  LOCUS   OF  AN  EQUATION  49 

concerned  chiefly  with  algebraic  curves  of  the  first  and 
second  degrees. 

37.   Construction  of  loci.      Discussion  of    equations.      The 

process  of  constructing  a  locus  by  plotting  separate  points, 
and  then  connecting  them  by  a  smooth  curve,  is  only  ap- 
proximate, and  is  long  and  tedious.  It  may  often  be  short- 
ened by  a  consideration  of  the  peculiarities  of  the  given 
equation,  such  as  symmetry,  the  limiting  values  of  the  vari- 
ables for  which  both  are  real,  etc.  Such  considerations  will 
often  show  the  general  form  and  limitations  of  the  curve  ; 
and,  taken  together,  they  constitute  a  discussion  of  the  equa- 
tion. 

The  points  where  a  locus  crosses  the  coordinate  axes  are 
almost  always  useful  ;  in  drawing  the  curve,  they  are  given 
by  their  distances  from  the  origin  along  the  respective  axes. 
These  distances  are  called  the  intercepts  of  the  curve. 

The  following  examples  may  serve  to  illustrate  these 
conceptions. 

(1)  Discussion  of  the  equation  3x  —  2y-\-12  =  0  [see  (3)  Art.  33]. 

Intercepts  :  if  x  =  0,  then  y  =  Q;  hence  the  ^/'intercept  is  6 
(see  Fig.  16) ;  it  y  =  0,  then  x  =  i]  hence  the  x-intercept  is  4. 

The  equation  may  be  written  :  x  =  ^  y  —  4,  which  shows  that  as  y 
increases  continuously  from  0  to  go  ,  a:  increases  continuously  from  —  4 
to  CO  ;  therefore  the  locus  passes  from  the  point  Pg  through  the  point  P^, 
and  then  recedes  indefinitely  from  both  axes  in  the  first  quadrant.  Writ- 
ten as  above,  the  equation  also  shows  that  as  y  decreases  from  0  to  —  co , 
X  also  decreases  from  —  4  to  —  co  ;  therefore  the  locus  passes  from  Pg  into 
the  third  quadrant,  receding  again  indefinitely  from  both  axes.  Since 
for  every  value  of  y,  x  takes  but  one  value  (i.e.,  each  value  of  y  corre- 
sponds to  but  one  point  on  the  curve),  therefore  the  locus  consists  of  a 
single  branch.  The  proof  that  the  locus  of  any  first-degree  equation,  in 
two  variables,  is  a  straight  line  is  given  in  Chap.  V. 

(2)  Discussion  of  the  equation  y'^  =  4x.     [See  (4)  Art.  33.] 
Intercepts  (see  Fig.  17)  :  if  a;  =  0,  then  y  =  0,  and  if  ?/  =  0,  then  x  —  0 ; 

TAN.   AN.  GEOM. 4 


50 


ANALYTIC  GEOMETRY 


[Ch.  III. 


hence  the  locus  cuts  each  axis  in  one  point  only,  and  that  point  is  the 
origin.  The  equation  may  be  written  in  the  form  ?/  =  ±  Vi  x,  which 
shows  that  if  x  be  negative  y  is  imaginary ;  hence  there  is  no  point  of 
this  locus  on  the  negative  side  of  the  ?/-axis. 

Again :  for  each  positive  value  of  x  there  are  two  real  values  of  y, 
numerically  equal,  but  opposite  in  sign ;  hence  this  locus  passes  through 
the  origin,  lies  wholly  in  the  first  and  fourth  quadrants,  and  is  symmetri- 
cal with  regard  to  the  rr-axis. 

The  equation  shows  also  that  x  may  have  any  positive  value,  however 

great,  and  that  y  increases  when  x  increases ;  these  facts  show  that  the 

locus  recedes  indefinitely  from  both  axes,  —  that  it  is  an  open  curve  of 

one  branch.     It  is  called  a  parabola  and  has  the  form  shown  in  Fig.  17. 

(3)  Discussion  of  the  equation  a;^  +  y^  _  q2 

Intercepts  :  if  a:  =  0,  then  y  =  ±  a,  and 
iJL  y  =  0,  then  x  =  ±a;  hence  for  each 
axis  there  are  two  intercepts,  each  of  length 
a,  and  on  opposite  sides  of  the  origin ; 
i.e.,  four  positions  of  the  tracing  point  are  : 
A=(a,  0),  A'  =  (-a,  0),  B=(0,  a),  and 
^'  =  (0,  -a). 

This  equation  may  also  be  written 
y  =  ±  Va^  —  x'^, 

which  shows  that  every  value  of  x  gives 
two  corresponding  values  of  y  which  are 
numerically  equal,  but  of  opposite  sign ; 
the  locus  is,  therefore,  symmetrical  with  regard  to  the  x-axis.  It  also 
shows  that,  corresponding  to  any  value  of  x  numerically  greater  than  a, 
y  is  imaginary;  the  tracing  point,  therefore,  does  not  move  further  from 
the  ?/-axis  than  ±  a,  i.e.,  further  than  the  points  A  and  A'.  Moreover, 
as  X  increases  from  0  to  a,  y  remains  real  and  changes  gradually  from 
+  a  to  0,  or  from  —a  to  0 ;  i.e.,  the  tracing  point  moves  continuously 
from  B  to  A,  or  from  B'  to  A. 

Again,  if  x  decreases  from  0  to  —  a,  y  remains  real  and  changes  con- 
tinuously from  +  a  to  0,  or  from  —  a  to  0 ;  i.e.,  the  tracing  point  moves 
continuously  from  B  to  A'  or  from  B  to  ^'. 

Similarly,  the  equation  may  be  written  x  =  ±  Va^  _  y'i^  which  shows 
that  the  curve  is  also  symmetrical  with  regard  to  the  ?/-axis,  and  that 
the  tracing  point  does  not  move  farther  than  ±  «  from  the  a:-axis. 

From  these  facts  it  follows  that  this  locus  is  a  closed  curve  of  only  one 
branch.  That  it  is  a  circle  of  radius  a,  with  its  center  at  the  origin,  will 
be  shown  in  Chap.  VII. 


37.] 


THE  LOCUS  OF  AN  EQUATION 


61 


(4)  Discussion  of  the  equation  y^  =  {x  —  2)  (x  —  3)  (a:  —  4). 
Intercepts :   if  a:  =  0,  then  y  is  imaginari/ ;  if  ?/  =  0,  then  a;  =  2,  3,  or 

4;  hence  the  locus  crosses  the  x-axes  at  the  three  points:  ^=(2,  0), 
5  =  (3,  0),  and  C=  (4,  0),  and 
it  does  not  cut  the  ?/-axis  at  all. 
Moreover,  since  y  is  imaginary  if 
X  is  negative,  the  locus  lies  wholly 
on  the  positive  side  of  the  ?/-axis. 
This  locus  is  symmetrical  with 
regard  to  the  ar-axis;  it  has  no 
point  nearer  to  the  ?/-axis  than 
A ;  between  A  and  B  it  consists 
of  a  closed  branch ;  and  it  has  no 
real  points  between  B  and  C,  but 
is  again  real  beyond  C.  The 
entire  locus  consists,  then,  of  a 
closed  oval,  and  of  an  open  branch 
which  recedes  indefinitely  from 
both  axes,  see  Fig.  22.  Fig.'^-^.  j: 

(5)  Discussion  of  the  equation  y  =  tan  x.  This  equation  has  already 
been  examined  in  (6)  Art.  33,  but  in  practice  it  may  be  much  more  simply 
plotted  by  the  following  method  : 

Describe  a  circle  with  unit  radius;  draw  the  diameter  ADC,  and  the 
lines  OB^,  OB^,  OB.,  •",  meeting  the  tangent  AT 
in  the  points  T^,  T^,  ^s^"-;  tlien  the  tangent  of 
the  angle  AOB^  is  M^B^^ :  OM^  =  AT^:  OA  (Art. 
14),  and,  since  OA  =1,  its  value  is  graphically  rep- 
resented by  A  Ty     So  also 

tan  ^0^2  =  -^2^2 '  OM^  =  AT,^:  0A=  AT^:1, 
and  may  be  graphically  represented  by  A  T^.  In 
the  same  way,  A  T.,  A  T^,  AT^,  •••  are  the  tangents 
of  the  angles  AOB^,  AOB^,  AOBq,  •••.  Again, 
since  angles  at  the  center  of  a  circle  are  propor- 
tional to  the  arcs  intercepted  by  their  sides,  A  T^, 
AT^,  •••  may  be  said  to  be  the  tangents  of  the 
i.e.,  AT^  =  tan  AB,,  A  T^  =  tan 
Therefore  the  coordinates  of  the  points 


arcs  AB^,  ABc^, 


AB,,- 

P^=(AB„  AT^),  P,  =  {AB.2,  AT^),"-  satisfy  the 
given  equation,  and  if  a  sufficient  number  of  points, 
whose  coordinates  are  thus  determined,  be  plotted, 
they  will  all  lie  on  a  curve  like  that  in  Fig.  19. 


52  ANALYTIC   GEOMETRY  [Cn.  III. 

From  what  has  just  been  said  it  is  clear  that  ?/  =  0  if  x  =  0,  hence  the 
curve  goes  through  the  origin ;  when  x  increases  continuously  from  0  to 
— ,  y  increases  continuously  from  0  to  oo,  but  when  x  increases  through  — , 
?/  passes  suddenly  from  +  oo  to  —  oo,  and  the  curve  is  discontinuous  for 
that  value  of  x.  So  also  when  x  increases  continuously  from  —  to  '— -, 
y  increases  continuously  from  —  oo  through  0  to  -f  oo,  and  is  again  dis- 
continuous  for  x  =  — .  The  locus  consists  of  an  infinite  number  of 
such  infinite,  but  continuous  branches,  separated  by  the  points  of  discon- 
tinuity for  which  x  =  ±—,  x  =  ±  ——,  x  =  ±  —,  •••. 

The  other  trigonometric  functions,  y  =  sin  x,  y  =  sec  x,  etc.,  can  all  be 
plotted  by  a  method  analogous  to  that  above. 

EXERCISES 

Construct  and  discuss  the  loci  of  the  following  equations : 


(cf .  Ex.  8,  p.  8.) 


38.  The  locus  of  an  equation  remains  unchanged:  (a)  by 
any  transposition  of  the  terms  of  the  equation ;  and  (P)  by 
multiplying  both  members  of  the  equation  by  any  finite  con- 
stant. 

(a)  If  in  any  equation  the  terms  are  transposed  from  one 
member  to  the  other  in  any  way  whatever,  the  locus  of  the 
equation  is  not  changed  thereby  ;  for  the  coordinates  of  all 
the  points  which  satisfied  the  equation  in  its  original  form, 
and  only  those  coordinates,  satisfy  it  after  the  transpositions 
are  made.      [See  Art.  35  (1).] 

(/3)  If  both  members  of  an  equation  are  multiplied  by  any 
finite  constant  ^,  its  locus  is  not  changed  thereby.  For  if 
the  terms  of  the  equation,  after  the  multiplication  has  been 
performed,  are  all  transposed  to  the  first  member,  that  mem- 
ber may  be  written  as  the  product  of  the  constant  k  and  a 


1. 

a;2      ?/2  _  ^ 

3. 

y  —  sec  X. 

7. 

V  =  sin  u. 

4       9 

4. 

x'^  —  ?/-=  or. 

8. 

a'2  M  f  =  0. 

2. 

^  +  ^  =  1. 
4       9 

5. 
6. 

x^-f^  0. 

9. 

?/-l_5-i 

y-2 

37-39.] . 


THE  LOCUS   OF  AN  EQUATION 


53 


factor  containing  the  variables.  This  product  will  vanish  if, 
and  only  if,  its  second  factor  vanishes  ;  but  this  factor  Avill 
vanish  if,  and  only  if,  the  variables  which  it  contains  are  the 
coordinates  of  points  on  the  locus  of  the  original  equation. 
Hence  the  coordinates  of  all  points  on  the  locus  of  the  ori- 
ginal equation,  and  only  those  coordinates,  satisfy  the  equation 
after  it  has  been  multiplied  by  k  ;  hence  the  locus  remains 
unchanged  if  its  equation  is  multiplied  by  a  finite  constant. 

39.  Points  of  intersection  of  two  loci.  Since  the  points  of 
intersection  of  two  loci  are  points  on  each  locus,  therefore 
the  coordinates  of  these  points  must  satisfy  each  of  the  two 
equations  ;  moreover,  the  coordinates  of  no  other  points  can 
satisfy  both  equations.  Hence,  to  find  the  coordinates  of  the 
points  of  intersection  of  two  curves,  it  is  only  necessary  to 
regard  their  equations  as  simultaneous  and  solve  for  the 
coordinates. 

E.g.,  Find  the  coordinates  of  the  points  of  intersection,  P^  and  Pg'  ^^ 
the  loci  oi  X  —  2  y  =  0,  and  y'^  =  x.  The  point  of  intersection  P^=  (i\,  y^) 
is  on  both  curves, 

.'.    Xj  —  2  y^  =  0,  Siud  y^  =  x^* 

Solving  these  tv^o  equations, 
x^  =  0,  or  4,  and  ?/j  =  0,  or  2 ; 

i.e.,  Pi=  (4,  2)  and  P^=  (0,  0)  are 
two  points,  the  coordinates  of  which 
satisfy  each  of  the  two  given  equa- 
tions ;  therefore  they  are  the  points 
of  intersection  of  the  loci  of  these 
equations. 

EXERCISES 
Find  the  points  of  intersection  of  the  following  pairs  of  curves : 
^7x-Uy-\-l=0,  ^      ^x-\-y  =  3. 


Fig.  31. 


1. 


Ix  +y~2  =  0. 


2. 


x-y 


*  If.  X  and  y  are  regarded  as  the  coordinates  of  the  point  of  intersection, 
the  subscripts  may  be  omitted  here. 


54  ANALYTIC  GEOMETRY  [Ch.  III. 

(y=Sx  +  2,  ^x  +  y  =  2a, 

^      (2y-5x  =  0,  9      (^'+y'  =  16, 

(  a;2  -  2^2  _  5,  ix       ^y  —  1. 

a:2+2/2  =  9,  10.    1^'=^^' 


j  a;2 


a:2  +  6  2'^  +  ^^  =  0.  •'^  ~    ' 

^  (p  =  9  COS  (45°-^), 

(?/2  =  4;9^,  12.     ^  /7r^^\      , 

^'     iy-x  =  0.  lpco.[-  +  0)  =  l. 

13.  Trace  carefully  the  above  loci;  by  measurement,  find  the  coordi- 
nates of  the  points  in  which  each  pair  intersect ;  and  compare  these 
results  with  those  already  obtained  by  computation. 

40.  Product  of  two  or  more  equations.  Given  two  or  more 
equations  with  their  second  members  zero  ;  *  the  product  of  their 
first  members^  equated  to  zero^  has  for  its  locus  the  combined 
loci  of  the  given  equations. 

This  follows  at  once  from  the  fundamental  relation  be- 
tween an  equation  and  its  locus  (see  Art.  35  (1)),  for  the 
new  equation  is  satisfied  by  the  coordinates  of  those  points 
which  make  one  of  its  factors  zero,  but  it  is  satisfied  by 
the  coordinates  of  no  other  points;  ^.6.,  this  new  equation 
is  satisfied  by  the  coordinates  of  points  that  lie  on  one  or 
another  of  the  loci  of  the  given  equations. 

The  following  example  illustrates  this  principle  in  the 
case  of  two  given  equations. 

Let  the  given  equations  be  : 
x^y  z=0     .     .      .     (1)  and  x-y  =  0     .     .     .     (2) 

*  If  equations  whose  second  members  are  not  zero  are  multiplied  together, 
member  by  member,  the  resulting  equation  is  not  satisfied  by  any  points 
of  the  loci  of  the  given  equations  except  those  in  which  they  intersect  each 
other ;  the  new  equation  therefore  represents  a  locus  through  the  points  of 
intersection  of  the  loci  of  the  given  equations. 


40.] 


THE  LOCUS   OF  AN   EQUATION 


55 


Y 

>? 

/ 

Pa         \ 

/ 

•                 \ 

y 

X 

\ 

\ 

^ 

\o 

Fig.  35. 


Equation  (1)  represents  the 
straight  line  CZ),  and  equation 
(2)  the  line  AB^  —  bisecting  re- 
spectively the  angles  between  the 
axes.  It  is  to  be  shown  that  the 
equation 

ix  +  y-){x~y-)=^  .     .    ,.     (3) 

(or,  what  is  the  same,  a:^—  ?/^=  0), 
formed  from  equations  (1)  and  (2), 
has  for  its  locus  both  these  lines. 

Proof.  If  P^  =  (iCj,  y-^  is  any  point  on  CD,  then  its  co- 
ordinates satisfy  equation  (1),  hence  x^-\-  yi  =  0,  and  there- 
fore (x^  +  ^j)  (x^  —  y^  =  0  ;  which  shows  that  P^  is  a  point 
of  the  locus  of  equation  (3).  But  since  P-^  was  any  point 
of  CP,  therefore  the  coordinates  of  every  point  on  QP  satisfy 
equation  (3);  i.e.,  all  points  of  CP  belong  to  the  locus  of 
equation  (3). 

In  the  same  way  it  is  shown  that  AP  belongs  to  the 
locus  of  equation  (3). 

Moreover,  if  P^^{x2^,  y^  be  any  point  not  on  AP  nor 
on  CP,  then  ^3  4- ^3  ^  0,  and  x^^  —  y^,^^  0,  hence 

i.e.,  Pg  does  not  belong  to  the  locus  of  equation  (3). 

Hence  the  locus  of  equation  (3)  contains  the  loci  of  equa- 
tions (1)  and  (2),  but  contains  no  other  points. 

The  above  theorem  may  be  stated  briefly  thus :  if  u,  v,  w, 
etc.,  be  any  functions  of  two  variables,  then  the  equation 
uvw  '  •••  =0  has  for  its  locus  the  combined  loci  of  the 
equations  u  =  0,  v  =  0,  w  =  0,  etc. 

Note.  When  possible,  factoring  the  first  member  of  an  equation, 
whose  second  member  is  zero,  simplifies  the  work  of  finding  the  locus  of 
the  given  equation. 


56  ANALYTIC   GEOMETRY  [Ch.  III. 

EXERCISES 

What  loci  are  represented  by  the  following  equations  V 

1.    xy  =  0.  2.   ^-^z=:0.  3.    'dx^  +  2xy-7x  =  0. 

^  4       9 

4.    5xy-2-2x''y  =  0.  5.    x2-2a;  +  l  =  0.        6.    (xHy^-4){y^-^x)=0. 


41.   Locus  represented  by  the  sum  of  two  equations.     Sup- 
pose the  equations 
2  ^  -  2)  =  0     .     .     .     (1),  and   j/2  -  a:  =  0     .     .     .     (2) 

are  given.      Their  loci   are   respectively  AB  and  DP^P^C 
(Art.  39),  and  it  is  required  to  find  the  locus  of  their  sum  ; 

^.e.,  of    2  ?/ —  X  +  ?/^  —  rr  =  0, 
or,  what  is  the  same  thing,  of 

./2+27/-2:r  =  0   .    .    .  ^3) 

The  locus  of  this  last  equa- 
tion passes  through  all  the 
points  in  which  AB  and 
DP^P^  C  intersect  each  other. 
For  let  P-^=(x-^^  y^  be  one  of 
these  points,  then  since  P^ 
lies  on  AB^  its  coordinates  satisfy  equation  (1);  z.e., 

2y^-x,  =  0;        .         .         .         (4) 

and  since  P^  lies  on  DP^P^C,  its  coordinates  satisfy  equa- 
tions (2);  z.e., 

^i^-^i  =  ^5        .         .         .         O'^) 

therefore,  by  adding  equations  (4)  and  (5), 

.       ^,2 +  2^1 -2:^1  =  0.      ...       (6) 

This  last  equation  proves  (Art.  35  (1))  that  P^=(x^,  y^) 
is  on  the  locus  of  equation  (3);  ^.e.,  the  locus  of  equation 
(3)  passes  through  P-^  =  (^x^,  y-^. 

Similar  reasoning  would  show  that  the  locus  of  equation 


Fig. 26. 


41.] 


THE  LOCUS   OF  AN  EQUATION 


57 


(3)  passes  through  every  other  point  in  which  the  loci  of 
equations  (1)  and  (2)  intersect  each  other. 

In  precisely  the  same  way  it  may  be  proved  generally  that 
the  locus  of  the  sum  of  tioo  equations  passes  through  all  the 
points  in  ivhich  the  loci  of  the  two  given  equations  intersect 
each  other. 

If  either  of  the  given  equations  (1)  or  (2)  had  been  multi- 
plied by  any  constant  factor  before  adding,  the  above  reason- 
ing would  still  have  led  to  the  same  conclusion  ;  in  fact, 
this  theorem  may  be  briefly,  and  more  generally,  stated  thus  : 
if  u  and  v  are  any  functions  of  the  two  variables  x  and  y,  and 
k  is  any  constant.,  then  the  locus  of 

u  -\-hv  =  0 

passes  through  every  point  of  intersection  of  the  loci  of 

u  =  0  and  V  =  0. 

For,  let  the  locus  of  the  equation  w  =  0  be  the  curve 
ABC^  the  locus  of  v  =  0  be  the  curve  BEF^  and  let 
Pi  =  (xi^  yi)  be  any  one  of 
the  points  in  which  these 
curves  intersect  each  other. 

Then  the  equation 

u  -{-  kv  =  0 

is  satisfied  by  the  coordi- 
nates of  the  point  Pi  = 
(^n  yi)->  because  if  these 
coordinates  be  substituted  for  x  and  y  in  the  functions  u  and 
V  they  must  make  both  these  functions  separately  equal  to 
zero.  Therefore  the  locus  of  u  -\-  kv  —  0  passes  through 
every  point  in  which  the  loci  oi  u  =  0  and  v  =  0  intersect 
each  other. 


Fig. 37 


58  ANALYTIC  GEOMETRY  [Ch.  III. 

EXERCISES 

1.  Verify  Art.  41  by  first  finding  the  coordinates  of  the  points  of 
intersection  of  the  loci  of  equations  (1)  and  (2),  and  then  substituting 
these  coordinates  in  equation  (3). 

2.  Find  the  equation  of  a  curve  that  passes  through  all  the  points  in 
which  the  following  pairs  of  curves  intersect : 

^""^    lx''  +  2x+i/  =  0.  i  "^f^^   \y  =  2  cos  X.  j 

3.  Find  the  equation  of  a  curve  through  all  the  points  common  to  the 

following  pairs  of  curves : 

^";  ly^  =  4:x.\  ^^^    (pcos^:-!.; 

Note.  It  is  to  be  observed  that  the  method  given  in  Art.  39,  for  find- 
ing the  point  of  intersection  of  two  curves,  is  an  application  of  the 
theorem  of  Art.  41.  For  the  process  of  solving  two  simultaneous  equa- 
tions, at  least  one  of  which  involves  two  variables,  consists  in  combining 
them  in  such  a  w^ay  as  to  obtain  two  simple  equations,  each  involving 
only  one  variable.  Now  each  of  these  simple  equations  represents  an 
elementary  locus,  —  one  or  more  straight  lines  parallel  to  the  axes,  if  the 
coordinates  are  Cartesian ;  circles  about  the  pole,  or  straight  lines  through 
the  pole,  if  the  coordinates  are  polar,  —  and  these  elementary  loci  deter- 
mine, i.e.,  pass  through,  the  points  of  intersection  of  the  original  loci. 
To  determine  the  points  of  intersection,  then,  of  two  loci,  the  original 
loci  are  replaced  by  simpler  ones  passing  through  the  same  common 
points.     E.g.,  the  points  of  intersection  of  the  loci  of  Art.  39, 

2y-x  =  0     .     .     .     (1),     and     y^  =  x,     .     .     .     (2) 

are  given  by  the  equations 

(y'^-x)  -(2y  -x)=0     and     (2  y  -  x)^  -  ^  (y^  -x)  =  0, 
that  is,  by  y^  —  2  y  =  0,     and     x^  —  4:  x  =  0, 

which  may  be  written 

y(y-2)  =  0     .     .     .     (3),  x(x-4.)  =  0.     ...     (4) 

But  the  locus  of  equation  (3)  is  a  pair  of  straight  lines  parallel  to  the 
a;-axis,  and  the  locus  of  equation  (4)  is  a  pair  of  straight  lines  parallel 
to  the  ?/-axis ;  and  these  loci  have  the  same  points  of  intersection  as  the 
loci  (1)  and  (2). 


41]  THE  LOCUS   OF  AN  EQUATION  59 

EXAMPLES  ON   CHAPTER   III 

1.  Are  the  points  (3,  9),  (4,  6),  and  (5,  5)  on  the  locus  of  3  x-\-2y  =  25'i 

2.  Is  the  point  l^,^\  on  the  locus  of  4  x^  +  9  ^/^  =  2  a^  ? 

3.  The  ordinate  of  a  certain  point  on  the  locus  of  x^  +  y'^  =  25  is  4  ; 
what  is  its  abscissa?     What  is  the  ordinate  if  the  abscissa  is  a'^? 

Find  by  the  method  of  Art.  39  where  the  following  loci  cut  the  axes 
of  X  and  y. 

4.  y  =  (x-2)(x-S).  5.    IQ  x^  +  9  y^  =  lU. 

6.  a;2  +  6  a;  +  2/2  =  4?/  4-  3. 

Find  by  the  method  of  Art.  39  where  the  following  loci  cut  the  polar 
axis  (or  initial  line). 

7.  p  =  4sin2^.  8.   p2  ^  a^  cos  2  ^. 

9.  The  two  loci —  =  1, — 1-^  =  1  intersect  in  four  points;  find 

4       9  4       9  F  » 

the  lengths  of  the  sides  and  of  the  diagonals  of  the  quadrilateral  formed 
by  these  points. 

10.  A  triangle  is  formed  by  the  points  of  intersection  of  the  loci  of 
x  -{■  y  =  a^  a:  —  2  2/  =  4a,  and  y  —  x  -\-1  a  —  ^.     Find  its  area. 

11.  Find  the  distance  between  the  points  of  intersection  of  the  curves 
3  a:  -  2  y  +  12  =  0,  and  a;2  +  ?/2  =  9. 

12.  Does  the  locus  of  ?/2  =  4  x  intersect  the  locus  of  2  a:  +  3  ?/  +  2  =  0? 

13.  Does  the  locus  of  x"^  —  ^y-\-^:  =  ^  cut  the  locus  of  a:2  +  2/2  =  l  ? 

14.  For  what  values  of  m  will  the  curves  a:2  +  ?/2  iz:  9  and  ?/  =  6  a:  +  m 
not  intersect?     (cf.  Art.  9.)     Trace  these  curves. 

15.  For  what  value  of  h  will  the  curves  ?/2  =  4x  and  y  —  x  -\-^  inter- 
sect in  two  distinct  points?  in  two  coincident  points?  in  two  imaginary 
points  (i.e.,  not  intersect)? 

16.  Find  those  two  values  of  c  for  which  the  points  of  intersection  of 
the  curves  ?/  =  2  a:  +  c  and  x"^  -\-  y^  —  25  are  coincident. 

17.  Find  the  equation  of  a  curve  which  passes  through  all  the  points 
of  intersection  of  xP-  +  1/2  —  25  and  y'^  =  ^x.  Test  the  correctness  of  the 
result  by  finding  the  coordinates  of  the  points  of  intersection  and  sub- 
stituting them  in  the  equation  just  found. 


60  ANALYTIC  GEOMETRY  [Ch.  Ill  41. 

18.  "Write  an  equation  which  shall  represent  the  combined  loci  of  (1), 
(2),  and  (3)  of  Art.  37. 

Discuss  and  construct  the  loci  of  the  equations : 

19.  (a-2  -  y-^)  {y  -  tan  x)  =  0.       22.    y   =  x\  25.  p  =  a^  cos  2  0. 

20.  x^  -  y^  =  0.                                 23.    y^  =  x^.  26.  p  =  3  0. 

21.  a;4  -  y^  =  0.                                 24.    y   =  10^  27.  p  =  a  sin  2  ^. 

28.    Show  that  the  following  pairs  of  curves  intersect  each  other  in 
two  coincident  points ;  i.e.,  are  tangent  to  each  other. 

3/2-  lOz-  6y-31  =0, 


(a)   . 


(9 


9^2  _  4^2  +  54:3;-  IQy  +  29  =  0, 
2  ?/  —  3  a;  +  5  =  0. 
29.    Find  the  points  of  intersection  of  the  curves 

^  +  f.=  1   and    ^-y-=l. 
25      9  25      9 


CHAPTER   IV 
THE   EQUATION  OF   A  LOCUS 

42.  The  equation  of  a  locus.  The  second  fundamental 
problem  of  analytic  geometry  is  the  reverse  of  the  first 
(cf.  Art.  31),  and  is  usually  more  difficult.  It  is  to  find, 
for  a  given  geometric  figure,  or  locus,  the  corresponding 
equation,  i.e.,  the  equation  which  shall  be  satisfied  by  the 
coordinates  of  every  point  of  the  given  locus,  and  which 
shall  not  be  satisfied  by  the  coordinates  of  any  other  point. 
The  geometric  figure  may  be  given  in  two  ways,  viz.  : 

(1)  As  a  figure  with  certain  known  properties ;   and 

(2)  As  the  path  of  a  point  which  moves  under  known 
conditions. 

In  the  latter  case  the  path  is  usually  unknown,  and  the 
complete  problem  is,  first  to  find  the  equation  of  the  path, 
and  then  from  this  equation  to  find  the  properties  of  the 
curve.     This  last  is  the  third  problem  mentioned  in  Art.  31. 

The  two  ways  by  which  a  locus  may  be  "  given  "  corre- 
spond to  the  two  conceptions  of  a  locus  mentioned  in  Art. 
35,  and  they  lead  to  somewhat  different  methods  of  obtaining 
the  equation.  The  first  method  may  be  exemplified  clearly, 
and  most  simply,  by  first  considering  the  familiar  cases  of 
the  straight  line  and  the  circle. 

43.  Equation  of  straight  line  through  two  given  points.* 

Let  Pj  =  (3,  2),  and  P^  =  (12,  5)  be  two  given  points ;   and 

*  See  also  Art.  51. 
61 


62 


ANALYTIC  GEOMETRY 


[Ch.  IV. 


let  P  =  (a:,  y)  be  ani/  other  point  on  the  line  through  P^ 


and  Pg. 


Draw  the  ordinates  M^P-^^  MP,  and  M^P^,  and  through 
Pj  draw  PjiV  parallel  to  the  a^-axis,  meeting  MP  in  R  and 


ilfgPa  in  i^2- 


T 

J^ 

R 

1 — 

--- 

1 
1 

-p- 

If 

0 

M, 

Fig.  28.^ 

M, 

■n 


The  triangles  P^RP  and  P^R^P^^  are  similar,  hence 


RP__P^R 


i^2^2 


Pji^g'  ^■^"  iff2P2  -  i^/jPi      Oifg  -  OM^ 


Substituting  for  MP,  OM,  M^P^,  OM^,  etc.,  their  values, 
this  equation  becomes 

?/-  2       rr-  3 


which  reduces  to 


5  _  2     12-3' 

3^  _ir-  3  =  0. 


(1) 


This  is  the  required  equation  of  the  straight  line  through 
Pj  and  Pg,  because  it  fulfills  both  the  requirements  of  the 
definition  [cf.  Art.  35  (1)];  ^•e.,  it  is  satisfied  by  the  coordi- 
nates of  any  (i.e.,  of  every)  point  of  this  line,  because  x,  y  are 
the  coordinates  of  any  such  point ;  and  it  i%  not  satisfied  by 
the  coordinates  of  any  point  which  is  not  on  this  line,  because 
the  corresponding  constructions  for  such  a  point  would  not 
give  similar  triangles,  and  hence  the  proportions  which  led 
to  this  equation  would  not  be  true. 

That  equation  (1)  is  not  satisfied  by  the  coordinates  of 


43-44.] 


THE  EQUATION   OF  A   LOCUS 


63 


any  point  not  on  the  line  through  Pj_  and  Pg  ^^7  ^^so  be 
seen  as  follows : 

let       ^3  =  (2:3,^3) 

be  any  point  not  on  the 


4   is   on   the   line  PiP^i 


its 


line  through  P^  and  P^^ 

the    ordinate  M^P^  will 

meet  P1P2  ^^  some  point 

P4  =  (x^,  y^,  for  which 

x^  =  x^   but    ?/4  =7^  y^.      Since    P 

coordinates  satisfy  equation  (1),  therefore 

3  ^4  -  ^4  -  3  =  0, 

.'.   3^3-2:3-3=5^0;*      [since  x^  =  x^  and  y^  ^  y^ 

hence  the  coordinates  of  Pg  do  not  satisfy  the  equation 

3^  —  a?  =  3. 

44.  Equation  of  straight  line  passing  through  given  point 
and  in  given  direction.!  Let  P^  =  (5,  4)  be  the  given  point, 
let  the  given  line  through  Pj  make  an  angle  of  30°  with  the 
a;-axis,  and  let  P  =  (2:,  y')  be  any  other  point  on  this  line. 

Draw  the  ordinates  M^P^  and  ifeTP,  and,  through  P^,  draw 
P^B,  parallel  to  the  a:-axis  to  meet  MP  in  R.     Then 

MP  -  M^P^ 


tan  RP^P  =  p  p 


OM  -  OM^ 


*  This  proof  shows  clearly  that  if  the  coordinates  of  any  point  on  the 
straight  line  through  Pi  and  P^  are  substituted  for  x  and  y  in  equation  (1) 
the  first  member  will  be  equal  to  zero  ;  if  the  coordinates  of  any  point  below 
this  line  are  so  substituted  the  first  member  will  be  negative  ;  and  if  the  coor- 
dinates of  any  point  above  this  line  are  so  substituted  the  first  jn ember  will  be 
positive.  This  line  may  then  be  regarded  as  the  boundary  which  separates 
that  part  of  the  plane  for  which  3?/  —  x  —  3  is  negative  from  the  part  for 
which  this  function  is  positive.  Because  of  this  fact  that  side  of  this  line  on 
which  P3  lies  may  be  called  the  negative  side,  and  the  other  the  positive  side. 

t  See  also  Art.  53. 


64 


ANALYTIC  GEOMETRY 


[Ch.  IV. 


Substituting  for  M^P^,  MP,  OM^,  OM,  and  angle  RP^P 
their  values,  and  remembering  that  tan  30°  =  — -  =  ^  V8, 
this  equation  becomes 


-U-A.    V 


i.e.,  x-VS^-5  +  4V3  =  0/ 


Fig.  30.— 


The  equation  just  found  is  satisfied  by  the  coordinates  of 
any  point  on  the  given  line,  but  is  not  satisfied  by  the  coor- 
dinates of  any  point  that  is  i^ot  on  this  line  (cf.  Art.  43); 
hence  it  is  the  equation  of  the  line  (cf.  Art.  35). 


45.  Equation  of  a  circle ;  polar  coordinates,  f  In  deriving 
this  equation,  let  polar  coordinates  be  employed,  merely  for 

variety,  and  let  the  pole  be  taken 
on  the  circumference,  with  a  di- 
ameter OA  extended  for  the  ini- 
tial line.  Let  P  ^  (p,  ^)  be  any 
point  on  the  circle,^  and  let  r  be 
the  radius  of  the  circle. 
j^iG^si^  Connect  P  and  A  by  a  straight 

*  The  positive  side  of  this  line  is  tliat  side  on  which  the  origin  lies  (cf. 
foot-note,  Art.  43). 

t  See  also  Art.  98. 

{Except  in  plane  geometry,  the  word  "circle"  is  employed  by  most 
writers  on  mathematics  to  mean  "circumference  of  a  circle."  It  will  be  so 
used  in  this  book. 


44-46.]  THE  EQUATION   OF  A   LOCUS  65 

line ;  then,  iu  triangle  A  OP,  angle  OP  A  is  a  right  angle, 
A  0P  =  e,  OP  =  p,  and  OP:OA  =  cos  0  ;  i.e., 

p  :  2r  =  cos  0  ; 
hence  p  =2r  cos 0.         .         .         .        (1) 

Equation  (1)  is  satisfied  by  the  polar  coordinates  of  every 
point  on  the  circle  ;  but  is  not  satisfied  by  the  coordinates 
of  a  point  Q  not  on  the  circle,  since  angle  AQO  is  not  a 
right  angle.  Therefore  Eq.  (1)  is  the  equation  of  this  circle 
(cf.  Art.  35). 

EXERCISES 

1.  Find  the  equation  of  the  straight  line  through  the  two  points  (1, 7) 
and  (6,  11)  ;  through  the  points  (~2,  5)  and  (3,  8).  Which  is  its  posi- 
tive side  of  these  lines  ? 

2.  Find  the  equation  of  the  straight  line  through  the  two  points  (2,  3) 
and  (-2,  -3).  Through  what  other  point  does  this  line  pass?  Does 
the  equation  show  this  fact  ? 

3.  Find  the  equation  of  the  straight  line  through  the  point  (5,  ~7), 
and  making  an  angle  of  45°  with  the  a:-axis ;  making  the  angle  —45°  with 
the  a:-axis. 

4.  Find  the  equation  of  the  line  through  the  point  (~6,  -2),  and 
making  the  angle  120°  with  the  a:-axis. 

5.  Construct  the  circle  whose  equation  is  p  =  10  cos  0. 

6.  With  rectangular  coordinates,  find  the  equation  of  the  circle  of 
radius  5,  which  passes  through  the  origin,  and  has  its  center  on  the 
X-axis.     Is  its  positive  side  outside  or  inside  ? 

46.  Equation  of  locus  traced  by  a  moving  point.  In  the 
problems  given  above,  the  geometric  figure  in  each  case  was 
completely  known  ;  and,  in  obtaining  its  equation,  use  was 
made  of  the  known  properties  of  similar  triangles,  triangles 
inscribed  in  a  semicircle,  and  trigonometric  functions.  In 
only  a  few  cases,  however,  is  the  curve  so  completely 
known  ;    in  a  large  class  of  important  problems,  the  curve 

TAN.   AN.  GEOM.  —  5 


66  ANALYTIC  GEOMETRY  [Ch.  IV. 

is  known  merely  as  the  path  traced  by  a  point  which  moves 
under  given  conditions  or  laws.  Such  a  curve,  for  instance, 
is  the  path  of  a  cannon  ball,  or  other  projectile,  moving 
under  the  influence  of  a  known  initial  force  and  the  force  of 
gravity.  Another  such  curve  is  that  in  which  iron  filings 
arrange  themselves  when  acted  upon  by  known  magnetic 
forces.  The  orbits  of  the  planets  and  other  astronomical 
bodies,  acting  under  the  influence  of  certain  centers  of  force, 
are  important  examples  of  this  class  of  "given  loci." 

In  such  problems  as  these,  the  method  used  in  Arts.  43  to  45, 
cannot,  in  general,  be  applied.  A  method  that  can  often  be 
employed,  after  the  construction  of  an  appropriate  figure,  is: 

(1)  From  the  figure,  express  the  known  law,  under  which 
the  point  moves,  by  means  of  an  equation  involving  geo- 
metric magnitudes  ;  this  equation  may  be  called  the  "  geo- 
metric equation." 

(2)  Replace  each  geometric  magnitude  by  its  equivalent 
algebraic  value,  expressed  in  terms  of  the  coordinates  of 
the  moving  point  and  given  constants  ;  then  simplify  this 
algebraic  equation,  and  the  result  is  the  desired  equation  of 
the  locus. 

47.    Equation  of  a  circle :    second  method.      To  illustrate 
this  second  method  of  finding  the  equation  of  a  locus,  con- 
sider the  cirx^le  as  the  path  traced  by  a 
point  which  moves  so  that  it  is  always 
at  a  given  constant  distance  from  a  fixed 
O  point.      From  this  definition,  find  its 

""^P    ^    equation. 

Let   C  =  (2>.    2)    be   the   sfiven   fixed 

point,  and  let  P  =(x^  ?/)  be  a  point  that 
moves  so  as  to  be  always  at  the  distance  21  from  C.     Then 

CP  =  j,     .     .     .      [geometric  equation] 


0 


46-48.]  THE  EQUATION  OF  A   LOCUS  67 

but         CP  =  V{x-Sy-{-(y  -2)2        (Art.  26,  [2]), 

.  • .   V(a:  —  8)2  4-  (^  _  2)2  =  A;  [algebraic  equation] 

^.^.,  (a; -3)2+ (y- 2)2=2/- ; 

hence  ^x^ +  4:i/^  -  24:x -I67/ -j- 21  =  0, 

which  is  the  required  equation. 

The  locus  of  this  equation  can  now  be  plotted  by  the 
methods  of  Art.  37,  and  its  form  and  limitations  can  be 
discussed  as  is  there  done  for  other  equations. 

EXERCISES 

1.  Find  the  equation  of  the  path  traced  by  a  point  which  moves  so 
that  it  is  always  at  the  distance  4  from  the  point  (5,  0).  Trace  the 
locus. 

2.  Find  the  equation  of  the  path  traced  by  a  point  which  moves  so 
that  it  is  always  equidistant  from  the  points  ("2,  3)  and  (7,  5)  (cf. 
Ex.  9,  p.  34). 

3.  A  line  is  3  units  long;  one  end  is  at  the  point  (-2,  3).  Find 
the  locus  of  the  other  end  (cf.  Ex.  8,  p.  34). 

4.  A  point  moves  so  as  to  be  always  equidistant  from  the  y-axis  and 
from  the  point  (4,  0).  Find  the  equation  of  its  path,  and  then  trace  and 
discuss  the  locus  from  its  equation. 

5.  A  point  moves  so  that  the  sum  of  its  distances  from  the  two  points 
(0,  Vo),  (0,  -VS)  is  always  equal  to  6.  Find  the  equation  of  the  locus 
traced  by  this  moving  point. 

6.  A  point  moves  so  that  the  difference  of  its  distances  from  the  two 
points  (0,  VH),  (0,  ~V5)  is  always  equal  to  2.  Find  the  equation  of  the 
locus  traced  by  this  moving  point. 

48.  The  conic  sections.  Of  the  innumerable  loci  which 
may  be  given  by  means  of  the  law  governing  the  motion  of 
the  generating  or  tracing  point,  there  is  one  class  of  par- 
ticular importance ;  and  it  is  to  the  study  of  this  important 
class  that  the  following  pages  will  be  chiefly  devoted.  These 
curves  are  traced  hy  a  point  which  moves  so  that  its  distance 


68  ANALYTIC  GEOMETRY  [Ch.  IV. 

from  a  fixed  point  always  hears  a  constant  ratio  to  its  distance 
from  a  fixed  straight  line.  These  curves  are  called  the  Conic 
Sections,  or  more  briefly  Conies,  because  they  can  be  obtained 
as  the  curves  of  intersection  of  planes  and  right  circular 
cones  ;  *  in  fact,  it  was  in  this  way  that  they  first  became 
known.  The  last  three  examples  just  given  belong  to  this 
class,  although  it  is  only  in  No.  4  that  this  fact  is  directly 
stated.  These  loci  are  the  parabola,  the  ellipse,  and  the 
hyperbola ;  it  will  be  shown  later  that  they  include  as  spe- 
cial cases  the  straight  line  and  the  circle,  f  They  are  of 
primary  importance  in  astronomy,  where  it  is  found  that  the 
orbit  of  a  heavenly  body  is  a  curve  of  this  kind. 

The  general  equation,  which  includes  all  of  these  curves, 

will  now  be  derived,  and  the  locus  briefly  discussed ;   in  a 

subsequent  chapter  will  be  given  a  detailed  study  of  the 

properties  of  these  curves  in  their  several  special  forms. 

(a)    The  equation  of  the  locus.     Let  F  be  the  fixed  point, 

—  the  focus  of  the  curve ;  B'D  the  fixed 

line, — the  directrix  of  the  curve ;  and  e 

the   given   ratio,  —  the   eccentricity   of 

the  curve. 

The  coordinate  axes  may  of  course 
be  chosen  as  is  most  convenient.  Let 
D'D  be  the  ?/-axis,  and  the  perpendicu- 
lar to  it  through  F,  i.e.,  the  line  OFX, 
be  the  cr-axis.  Let  P  =  (a;,  ?/)  be  any  position  of  the  generat- 
ing point,  and  let  OF,  the  fixed  distance  of  the  focus  from 
the  directrix,  be  denoted  by  k ;  then  the  coordinates  of  the 
focus  are  (k,  0).  Connect  F  and  P,  and  through  F  draw 
LP  perpendicular  to  the  directrix. 

Then  FP  :  LP  =  e,     [geometric  equation] 


Y 

D 

L 

--»¥* 

T^ 

/ 

0 

/ 

X 

F 

1 
D 

Fig.  33. 

*  See  Note  D,  Appendix.  t  See  Note  C,  Appendix. 


48.]  THE  EQUATION   OF  A   LOCUS  69 


but  FP  =  ^{x  -  kf  +  /     (Art.  26), 

and  LP  =  x  ;     [algebraic  equivalents] 

hence  V(a;  —  k)"^  +  y'^  =  ex; 

i.e.,  (l-e^)3^-\-y^-2kx  +  k^  =  0,     .     .      .      (1) 

whicli  is  the  equation  of  the  given  locus. 

This  equation  is  of  the  second  degree ;  in  a  later  chapter 
it  will  be  shown  that  every  equation  of  the  second  degree 
between  two  variables  represents  a  conic  section.  On  this 
account  it  is  often  spoken  of  as  the  "second  degree  curve." 

(6)    Discussion  of  equation  (1). 

If  a;  =  0,  then  y=±k  V—  1,  which  shows  that  this  curve 
does  not  intersect  the  ?/-axis  as  here  chosen;  ^.e.,  a  conic 
does  not  intersect  its  directrix. 

If  3/  =  0,  then  (1  -  e2)^2  _  2  kx -{- k"^  =  0, 

whence  x  = ,  or  a;  =  :; ,        .       .       .       (2) 

1  -\-  e  1  —  e 

i.e.,  a  conic  meets  the  line  drawn  through  the  focus  and  per- 
pendicular to  the  directrix  (the  2:-axis  as  here  chosen)  in 

k 
two  points  whose  distances  from  the  directrix  are  and 

k         .         .     .  1+^ 

I  _  respectively ;  these  points  are  called  the  vertices  of  the 
conic. 

Equation  (1)  shows  that  for  every  value  of  x,  the  two 
corresponding  values  of  y  are  numerically  equal  but  of 
opposite  signs,  hence  the  conic  is  symmetrical  with  regard 
to  the  a;-axis  as  here  chosen.  For  this  reason  the  line 
drawn  through  the  focus  of  a  conic  and  perpendicular  to 
the  directrix  is  called  the  principal  axis  of  the  conic. 

The  form  of  the  locus  of  equation  (1)  depends  upon  the 
value  of  the  eccentricity  (e)  ;   if  e  =  1,  the  conic  is  called  a 


70 


ANALYTIC  GEOMETRY 


[Ch.  IV. 


parabola ;    if  e  <  1,  an  ellipse ;    and  if  e  >  1,  an  hyperbola. 

Each  of  these  cases  will  now  be  separately  considered. 


(1)   The  parabola^ 


Fig.  34. 


e  =  l.  If  e  =  l,  then  FF:LF  =  1, 
^.e.,  FP  =  LP  for  every  position  of 
the  tracing  point,*  hence  the  curve 
passes  through  J.,  —  the  point  mid- 
way between  0  and  F^  —  but  does  not 
again  cross  the  principal  axis  (cf. 
also  equations  (2),  above). 

Moreover,  when  e  =  1,  equation  (1) 
becomes 


I.e.. 


/  -  2  ^rc  +  ^2  ^  0, 
/  =  2  klx  —  -J, 


(3) 


which  is  the  equation  of  the  parabola,  the  coordinate  axes 
being  the  principal  axis  of  the  curve  and  the  directrix. 
Equation  (3)  shows  that  there  is  no  point  of  this  parabola 

for  which  a^<-,  and  also  that  y  changes  from  0  to   ± go 

when  X  increases  from  -  to  oo ;  hence  the  parabola  recedes 

indefinitely  from  both  axes  in  the  first  and  fourth  quadrants. 

Its  form  is  given  in  Fig.  34. 

(2)   The  ellipse^  e  <1.     Equation  (1)  may  be  written  in 

the  form 

k  \f  h 


^2=  (1-^2) 


—  X 


X 


1  +  e 


(4)t 


*  This  property  enables  one  to  construct  any  number  of  points  lying  on  the 
parabola,  thus :  with  F  as  center,  and  any  radius  not  less  than  \0F^  describe 
a  circle,  then  draw  a  line  parallel  to  OY  and  at  a  distance  from  it  equal  to 
the  chosen  radius ;  the  points  in  which  this  line  cuts  the  circle  are  points  on 
the  parabola.  Other  points  can  be  located  in  the  same  way.  See  also  Note 
B,  Appendix. 

t  Equation  (4)  enables  one  to  construct  any  number  of  points  on  the 


48.] 


THE  EQUATION   OF  A    LOCUS 


71 


which  shows,  e  being  less  than  1,  that  ?/  is  imaginary  for  all 
values  of  x  except  those  which  satisfy  the  condition 


k     _     -     Jc 

X 


l^e<     <\-e' 

hence  the  ellipse  lies  wholly  on  the  positive  side  of  its  direc- 
trix, and  between  two  lines  which  are  parallel  to  the  directrix 

Y 

p 


and  distant  from  it 


and 


l-\-e  1-e 

tion  (4)  shows  that  as  x  increases  from 


respectively.      Equa- 
k       .         k 


1  +  e 


to 


^ 


are 


ellipse.    E.g.  ,\etx=  OM ;  then  the  factors  (x ]  and  ( x] 

\        1  +  e/  \l-e       J 

the  two  segments  AM 


and  MA'  of  the  line 
AA'^  and  geometri- 
cally their  product 
equals  the  square  of 
the  ordinate  31Q  of 
the  semicircle  of  which 
A  A'  is  the  diameter. 
If  now  the  point  P  on 
MQ  be  so  constructed 
that  MP  =  Vl  -  e2  •  MQ,  then  P  is  a  point  on  the  ellipse  whose  equation  is 
(4)  above. 

Similarly,  any  number  of  points  on  the  curve  can  be  constructed.  This 
method  shows  also  that  the  ordinates  of  an  ellipse  are  less  than,  but  in  a  con- 
stant ratio  to,  the  corresponding  ordinates  of  the  circle  of  which  the  diameter 
is  the  line  joining  the  vertices  of  the  ellipse.     See  also  Note  B,  Appendix. 


72 


increases  from  0  to 


k 


ANALYTIC  GEOMETRY 

ek 


[Ch.  IV. 


Vl  -  e2  V 


[which  value  it  reaches  when 


X  = J  and  then  decreases  again  to  0.     The  form  of  the 

curve    is    therefore    as    shown    in    Fig.   35,    where    0F=  k, 
OA  =  J^,   00  =  ^-^.   OA  =  J^,  and  CB  =^       ^^ 


1+e 


1-e^ 


1-e 


Vl-e=i 


(3)   The  hyperhola^    e>l.       Equation    (1)    may  also   be 
written  in  the  form 


^2  —  ^g2  _  l^(x  — 


k 


1  +  e 


X  — 


k 


1-e 


(5) 


which,  when  e>l.  shows  that  ?/  is  imaginary  for  all  values 

of  X  between  x  = and  x  = ,  and  that  y  is  real  for 

1+e  1-e  ^ 

all  other  values  of  x.     Equation  (5)  also  shows  that,  as  x 


increases  from 


k 


to  GO,  ?/  changes  from  0  to  ±co,  and 
k 


1  +  e 

that,  as  x  decreases  from 

1  —  e 

The   form  of   the  curve   is   therefore    as    show^n  in 


to  —  GO,  ?/  changes  from  0  to 


±  00. 


k 


k 


k 


Fig.  37,  where  OA  =  -^^^—  and  OA'  = 

^  1  +  e  1-e  e-1 

Although  these  three  curves  differ  so  widely  in  form,  they 

are  really  very  closely  related  as  will  be  further  shown  in 

Chap.  XII,  and  in  Note  D  of  the  Appendix. 


48-49.]  THE  EQUATION   OF  A   LOCUS  73 

49.  The  use  of  curves  in  applied  mathematics.*  In  Chap- 
ter III  it  was  shown  that  whenever  the  relation  between  two 
variables,  whose  values  depend  upon  each  other,  can  be  defi- 
nitely stated,  ^.e.,  when  the  variables  can  be  connected  by 
an  equation,  then  the  geometric  or  graphic  representation  of 
this  relation  is  given  by  means  of  a  curve.  Such  a  curve 
often  gives  at  a  glance,  information  which  would  otherwise 
require  considerable  computation  to  secure ;  and  in  many 
cases  it  brings  out  facts  of  peculiar  interest  and  importance 
which  might  otherwise  escape  notice. 

The  use  of  graphic  methods  in  the  study  of  physics  and 
engineering,  as  well  as  in  statistics  and  many  other  branches 
of  investigation,  is  already  extensive  and  is  rapidly  increas- 
ing. Under  the  name  "  graphic  methods "  there  are  in- 
cluded, however,  not  only  such  examples  as  those  already 
given,  where  the  equation  connecting  the  variables  is  known, 
but  also  those  where  no  such  equation  can  be  found  ;  in 
these  latter  cases  the  curves  constitute  almost  the  only  prac- 
tical way  of  studying  the  relations  involved. 

As  a  simple  example  of  this  kind,  suppose  the  temperature, 
of  a  patient  to  be  accurately  observed  at  intervals  of  one  hour ; 
if  the  numbers  representing  the  hours,  i.e.^  1,  2,  3,  •••  are 
taken  as  abscissas,  and  the  corresponding  numerical  values  of 
the  temperatures  be  taken  as  ordinates,  then  a  smooth  curve 
drawn  through  the  points  so  determined  will  express  graphi- 
cally the  variation  of  the  temperature  of  this  patient  with 
the  time.  This  curve  will  also  show  to  the  physician  what 
was  the  greatest  and  least  temperature  during  the  inter- 
val of  the  observations,  as  well  as  the  time  when  each  of 


*  For  most  of  the  suggestions  in  this  article,  and  in  the  examples  that 
follow  it,  the  authors  are  indebted  to  Mr.  J.  S.  Shearer  of  the  Department 
of  Physics  of  Cornell  University 


74  ANALYTIC  GEOMETRY  [Ch.  IV. 

these  was  attained.  In  this  problem  the  curve  gives  no 
new  information,  but  it  presents  in  a  much  more  concise 
and  forcible  form  the  information  given  by  the  tabulated 
numbers. 

Again,  if  the  distances  passed  over  by  a  train  in  successive 
minutes  during  the  run  between  two  stations  are  taken  as 
ordinates,  and  the  corresponding  number  of  minutes  since 
starting,  as  abscissas,  a  smooth  curve  drawn  through  the 
points  so  determined  will  show  at  a  glance,  to  an  experi- 
enced eye,  where  and  when  additional  steam  was  turned  into 
the  cylinders,  brakes  applied,  heavy  grades  encountered,  etc., 
etc. 

In  all  such  cases  the  coordinates  of  the  points  are  taken  to 
represent  the  numerical  values  of  related  quantities,  such  as 
time,  length,  weight,  velocity,  current,  temperature,  etc.,  and 
the  curve  through  the  points  so  determined  usually  gives,  to 
an  experienced  person,  all  the  information  concerning  the 
relations  involved  that  is  of  practical  importance.  It  is 
in  the  study  of  such  curves  that  much  of  the  value  of  train- 
ing in  analytic  geometry  becomes  apparent  to  the  physicist 
and  the  engineer.  The  student  should  early  learn  to  trans- 
late physical  laws  into  graphic  forms,  and  he  should  give 
careful  attention  to  the  interpretation  of  all  changes  of  form, 
intercepts,  intersections,  etc.,  of  such  curves. 

EXERCISES 

1.  In  simple  interest  if  jo  =  principal,  t=time,  ?'  =  rate,  and  a  =  amonnt, 
then  a  =p  (1  +  rt).  K  now  particular  numerical  values  are  given  to 
p  and  r,  and  if  the  values  of  the  variable  a  be  taken  as  ordinates,  and 
the  corresponding  values  of  t  as  abscissas,  then  the  locus  of  this  equa- 
tion may  be  drawn.  Draw  this  locus.  What  line  in  the  figure  repre- 
sents the  principal?  What  feature  of  the  curve  depends  upon  the  rate 
per  cent  ?    Interpret  the  intercepts  on  the  axes. 


49.] 


THE  EQUATION    OF  A   LOCUS 


75 


2.  Give  to  p  and  r  in  exercise  1  different  values  and,  with  the  same 
axes,  draw  the  corresponding  locus.  How  do  these  loci  differ?  What 
does  their  point  of  intersection  mean  ? 

3.  With  the  same  axes  as  before  draw  the  curve  for  which  interest  and 
time  are  the  coordinates;  how  is  it  related  to  the  curves  of  exercises 
1  and  2? 

4.  Draw  and  discuss  the  curve  showing  the  relation  between  amount^ 
principal,  rate,  and  time  in  the  case  of  compound  interest. 

(a)  When  interest  is  compounded  annually. 
(^)  When  interest  is  compounded  quarterly, 
(y)  When  interest  is  comx^ounded  instantaneously. 

5.  A  wage  earner  has  akeady  been  working  10  days  at  |1.50  per  day, 
and  continues  to  do  so  20  days  longer,  after  which  he  is  idle  during  8  days ; 
he  then  works  14  days  more  at  the  same  wages,  after  which  his  employer 
raises  his  wages  to  .*$2.50  per  day  for  the  next  20  days :  using  the  amounts 
earned  as  ordinates,  and  the  time  (in  days)  as  abscissas,  draw  carefully 
the  broken  line  which  states  the  above  facts. 

What  modification  of  the  drawing  would  be  necessary  to  show  that 
the  wage  earner  forfeited  50  cents  per  day  during  his  idleness  ? 

6.  The  following  table  shows  the  production  of  steel  in  Great  Britain 
and  the  United  States  from  1878  to  1891.* 


U.S. 

G.B. 

U.S. 

G.B. 

1878  .  . 

7.3  (100,000 
long  tons) 

10.6  (100,000 

long  tons) 

1885  .  . 

17.1 

19.7 

1879  .  . 

9.3 

10.9 

1886  .  . 

25.6 

23.4 

1880  .  . 

12.5 

13.7 

1887  .  . 

33.4 

31.5 

1881  .  . 

15.9 

18.6 

1888  .  . 

29.0 

34.0 

1882  .  . 

17.4 

21.9 

1889  .  . 

33.8 

36.7 

1883  .  . 

16.7 

20.9 

1890  .  . 

42.8 

36.8 

1881  .  . 

15.5 

18.5 

1891  .  . 

39.0 

32.5 

Using  time  (in  years)  as  abscissas,  and  quantity  of  steel  produced 
(100,000  tons  per  unit)  as  ordinates,  the  separate  points  represented  by 


*  Taken  by  permission  from  Lambert's  Analytic  Geometry. 


76 


ANALYTIC   GEOMETRY 


[Ch.  IV. 


the  table  have  been  plotted  (Fig.  38)  and  then  joined  by  straight  lines, 
dotted  for  Great  Britain  and  full  for  the  United  States.* 
Interpret  fully  the  figure. 
45 


40 
35 
30 
25 
20 
15 
10 
5 


/ 

\ 

.y 

- 

/ 

\ 

// 

^  / 

N 

f 

,y 

^ 

/y 

/ 

--^ 

' 

r^ 

^ 

/^ 

1878      79     '80 


"81      '83      '83      '84      '85 
Fig.  38. 


'87      '88 


'90      '91 


7.  Exhibit  graphically  the  mformation  contained  in  the  following 
table  on  the  expense  of  moving  freight  per  "ton-mile"  on  N.  Y.  C.  & 
H.  R.  R.  R.  from  1866  to  1893. 


1866 
1867 

1868 
1869 
1870 
1871 

1872 


2.16^ 

1.95 

1.80 

1.40 

1.15 

1.01 

1.13 


1873 
1874 

1875 
1876 
1877 
1878 
1879 


1.03^ 
.98 
.90 
.71 
.70 
.60 
.55 


1880 
1881 
1882 
1883 
1884 
1885 
1886 


.54^ 

.56 

.60 

.68 

.62 

.54 

.53 


1887 
1888 
1889 
1890 
1891 
1892 
1893 


.56;* 

.59 

.57 

.54 

.57 

.54 

.54 


8.  The  following  table  gives  the  population  of  the  countries  named 
between  1810  and  1896  :  f 

*  In  the  figure  the  linear  unit  on  the  2C-axis  is  5  times  as  long  as  the  linear 
unit  on  the  ?/-axis.  It  will,  however,  be  noticed  that  the  essential  feature  of 
a  system  of  coordinates,  the  "one-to-one  correspondence"  of  the  symbol 
(x,  y)  and  the  points  of  a  plane,  is  not  disturbed  by  using  different  scales  for 
ordinates  and  abscissas. 

t  The  authors  are  indebted  to  Professor  W.  F.  Willcox  of  Cornell  Univer- 
sity for  these  data,  which  are  compiled  from  the  Statesman'' s  Year  Book  for 
1897,  and  from  Statistik  des  Deutschen  Beichs,  Bd.  44,  1892. 


49.] 


THE  EQUATION   OF  A   LOCUS 


77 


British  Isles 

Lands  now  included  in  the 
German  Empire 

Year 

Population 

Year                       Population 

1801 

15,896,000 

1816 

24,831,000 

1811 

17,908,000 

1837 

31,540,000 

1821 

20,894,000 

1847 

34,753,000 

1831 

24,029,000 

1856 

36,130,000 

1841 

26,709,000 

1865 

39,399,000 

1851 

27,369,000 

1872 

41,028,000 

1861 

28,927,000 

1876 

42,775,000 

1871 

31,485,000 

1885 

46,856,000 

1881 

34,885,000 

1895 

52,280,000 

1891 

37,733,000 

France 

Ireland 

United  States 

Year 

Population 

Year 

Population 

Year 

Population 

1821 

30,462,000 

1811 

5,938,000 

1810 

7,240,000 

1841 

34,230,000 

1821 

6,802,000 

1820 

9,634,000 

1861 

37,386,000 

1831 

7,767,000 

1830 

12,866,000 

1866 

38,067,000 

1841 

8,175,000 

1840 

17,069,000 

1872 

36,103,000 

1851 

6,552,000 

1850 

23,192,000 

1876 

36,906,000 

1861 

5,799,000 

1860 

31,443,000 

1881 

37,672,000 

1871 

5,412,000 

1870 

38,558,000 

1886 

38,219,000 

1881 

5,175,000 

1880 

50,156,000 

1891 

38,343,000 

1891 

4,705,000 

1890 

62,622,000 

1896 

38,518,000 

Employing  the  number  of  years  as  abscissas,  and  the  population 
(500,000  per  unit, — numbers  at  left  of  figure  represent  millions)  as  ordi- 
nates,  the  separate  points  represented  by  the  above  table  have  been 
plotted  (Fig.  39)  and  then  joined  by  straight  lines.  The  figure  gives  all 
the  information  contained  in  the  tabulated  results,  besides  showing  at  a 
glance  the  relative  population  of  the  different  countries  at  any  given 
time.  The  student  may  account  historically  for  the  abrupt  fall  in  the 
line  representing  the  population  of  France;  and  for  the  gradual  down- 
ward tendency  in  the  line  representing  the  population  of  Ireland. 


78 


ANALYTIC  GEOMETRY 


[Ch.  IV. 


49.]  THE  EQUATION   OF  A   LOCUS  79 

EXAMPLES  ON   CHAPTER   IV 

1.  Find  the  equations  of  the  sides  of  the  triangle  whose  vertices  are 
the  points  (2,  3),  (4,  -5),  (3,  -6)  (cf.  Art.  43).  Test  the  resulting 
equations  by  substitution  of  the  given  coordinates. 

2.  Find  the  equations  of  the  sides  of  the  square  whose  vertices  are 
(0,  -1),  (2,  1),  (0,  3),  (-2,  1).  Compare  the  equations  of  the  parallel 
sides ;  of  perpendicular  sides. 

3.  Find  the  coordinates  of  the  center  of  the  square  in  Ex.  2.  Then 
find  the  radius  of  the  circumscribed  circle,  and  (Art.  47)  the  equation  of 
that  circle.  Test  the  result  by  finding  the  coordinates  of  the  points  of 
intersection  of  one  of  the. sides  with  circle  (Art.  39). 

4.  Find  the  equation  of  the  path  traced  by  a  point  which  is  always 
equidistant  from  the  points 

(a)     (2,  0)  and  (0, -2)  ;     (jS)     (3,  2)  and  (6,  6)  ; 

(y)     (<^  +  &,  «  —  &)  and  {a  —  b,  a  +  b). 

5.  A  point  moves  so  that  its  ordinate  always  exceeds  f  of  its  abscissa 
by  6.     Find  the  equation  of  its  locus,  and  trace  the  curve. 

6.  A  point  moves  so  thut  the  square  of  its  ordinate  is  always  4  times 
its  abscissa.     Find  the  equation  of  its  locus  and  trace  the  curve. 

7.  Find  the  equation  of  the  locus  of  a  point  which  moves  so  that  the 
sum  of  its  distances  from  the  points  (1,  3)  and  (4,  2)  is  always  5.  Trace 
and  discuss  the  curve. 

8.  Find  the  equation  of  the  locus  of  the  point  in  example  7,  if  the 
difference  of  its  distances  from  the  fixed  points  is  always  2. 

9.  Express  by  a  single  equation  the  fact  that  a  point  moves  so  that 
its  distance  from  the  a:-axis  is  always  numerically  3  times  its  distance 
from  the  ?/-axis. 

10.  A  point  moves  so  that  the  square  of  its  distance  from  the  point 
(a,  0)  is  4  times  its  ordinate.  Find  the  equation  of  its  locus,  and  trace 
the  curve. 

11.  A  point  moves  so  that  its  distance  from  the  x-axis  is  J  of  its  dis- 
tance from  the  origin.  Find  the  equation  of  its  locus,  and  trace  the 
curve. 

12.  A  point  moves  so  that  the  difference  of  the  squares  of  its  dis- 
tances from  the  points  (1,  3)  and  (4,  2)  is  5.  Find  the  equation  of  its 
locus  and  trace  the  curve. 


80  ANALYTIC  GEOMETRY  [Ch.  IV.  49. 

13.  Solve  example  12  if  the  word  "sum"  is  substituted  for  "differ- 
ence." 

14.  Let  A  =  (a,  0),  B  =  (b,  0),  and  A'  =  {-a,  0)  be  three  fixed  points; 
find  the  equation  of  the  locus  of  the  point  F  =  (x,  y)  which  moves  so 
that  PB'  +  PA^  =  2  PA'\ 

15.  A  point  moves  so  that  ^  of  its  abscissa  exceeds  ^  of  its  ordinate 
by  1.     Find  the  equation  of  its  locus  and  trace  the  curve. 

16.  Find  the  equation  of  the  locus  of  a  point  that  is  always  equi- 
distant from  the  points  (~3, 4)  and  (5,  3);  from  the  points  (~3, 4)  and 
(2,  0).  By  means  of  these  two  equations  find  the  coordinates  of  the 
point  that  is  equidistant  from  the  three  given  points. 

17.  Let  A  =  (-l,  3),  B=(-S,  -3),  C  =  (l,  2),  Z)  =  (2,  2)  be  four 
fixed  points,  and  let  P=(x,  y)  be  a  point  that  moves  subject  to  the  con- 
dition that  the  triangles  PAB  and  PCD  are  always  equal  in  area;  find 
the  equation  of  the  locus  of  P. 

18.  If  the  area  of  a  triangle  is  25  and  two  of  its  vertices  are  (5,  -  6) 
and  ("3,  4),  find  the  equation  of  the  locus  of  the  third  vertex. 

19.  A  point  moves  so. that  its  distance  from  the  pole  is  numerically 
equal  to  the  tangent  of  the  angle  which  the  straight  line  joining  it  to  the 
origin  makes  with  the  initial  line.  Find  the  polar  equation  of  its  locus 
and  plot  the  figure. 


CHAPTER   V 

THE  STRAIGHT  LINE.     EQUATION  OF  FIRST  DEGREE 

Ax  +  By  +  C  =  0 

50.  In  Chapter  III  it  was  shown  that  to  every  equation 
between  two  variables  there  corresponds  a  definite  geometric 
locus,  and  in  Chapter  IV  it  was  shown  that  if  the  geometric 
locus  be  given,  its  equation  may  be  found.  -It  still  remains 
to  exhibit  in  greater  detail  some  of  the  more  elementary  loci 
and  their  equations,  and  to  apply  analytic  methods  to  the 
study  of  the  properties  of  these  curves.  Since  the  straight 
line  is  a  simple  locus,  and  one  whose  properties  are  already 
well  understood  by  the  student,  its  equation  will  be  ex- 
amined first. 

In  studying  the  straight  line,  as  well  as  the  circle  and 
other  second  degree  curves,  to  be  taken  up  in  later  chapters, 
it  will  be  found  best  first  to  obtain  the  simplest  equation 
which  represents  the  locus,  and  to  study  the  properties  of 
the  curve  from  that  simple  or  standard  equation.  Then  it 
remains  to  find  methods  for  reducing  to  this  standard  form 
any  other  equation  that  represents  the  same  locus. 

51.  Equation  of  straight  line  through  two  given  points.     A 

numerical  example  of  the  equation  of  tlie  line  through  two 
fixed  points  has  already  been  given  in  Art.  43  ;  in  the  pres- 
ent article  the  equation  of  a  straight  line  tlirough  an^  two 
given  points  will  be  derived ;  the  method,  however,  will  be 
precisely  the  same  as  that  already  employed  in  the  numerical 
example. 

TAN.  AN.  GEOM.  —  6  81 


82 


ANALYTIC  GEOMETRY 


[Ch.  V. 


Let  the  two  given  fixed  points  be  Fi=(^xi,  y{)  and  P,^ 
(x2,  y-^^  and  let  P=(rr,  ?/)  be  any  other  point  on  the  line 
through  Pi  and  P^.     Draw  the  ordinates  MiP^^  -^2-^*21  ^^^ 


M 

Fig.  40.— 


rrQ.40.^ 


MP  \  also  through  P^  draw  P1R2  parallel  to  the  a;-axis,  and 
meeting  MP  in  R  and  M2P2  in  R2.  Then  the  triangles 
PiRP  and  P1R2P2  are  similar  ; 

RP__P^R      .        MP-M,P,  _  OM-  OM^ 

R2P2~  PiR2     ''""   M2P2-M,P,~  OM2-O3I,' 

Substituting  in  this  last  equation  the  coordinates  of  Pi, 

P2,  and  P,  it  becomes 

y  -y\  ^  a?  -  a?! 

^2  -  2/1     a?2  -  a?! '        •         •         *  "-J 

and  since  P  =  (x^  y)  is  a^i?/  point  on  the  line  through  Pi  and 
P2,  therefore  equation  [9]  is  satisfied  by  the  coordinates  of 
every  point  on  this  line.  That  equation  [9]  is  not  satisfied 
by  the  coordinates  of  any  point  except  such  as  are  on  the 
line  P1P2  may  be  proved  as  was  done  in  Art.  43. 

Equation  [9]  then  fulfills  both  requirements  of  the  defi- 
nition in  (1)  of  Art.  35,  and  is  therefore  the  equation  of 
the  straight  line  through  the  two  points  (a^i,  y{)  and  (2^2?  ^2)  • 
This  equation  will  be  frequently  needed  and  will  be  referred 
to  as  a  standard  form ;  it  should  be  committed  to  memory.* 


*  Throughout  this  book  the  more  important  formulas  are  printed  in  bold- 
faced type ;  they  should  be  committed  to  memory  by  the  learner. 


51-52.] 


THE  STRAIGHT  LINE 


83 


52.   Equation  of   straight  line  in  terms  of  the   intercepts 
which  it  makes  on  the  coordinate  axes.     If  the  two  given 


^J 

Y 

B 

"V. 

\^ 

0 

^^' 

^^ 

\^ 

X 

Fig.  43. 

4^ 

^M 

points  in  Art.  51  are  those  in  which  the  line  cuts  the  axes 

of  coordinates,  i.e.,  A=(a,  0)  and  B  =  (0,  6)  (Fig.  41),  then 

equation  [9]  becomes 

y  —  0  _x 

b-0~0 


a 


a 


that  is, 


^  +  y 

a      h 


1, 


[10] 


where  a  and  h  are  the  intercepts  which  the  line  cuts  from 
the  axes. 

This  is  another  standard  form  of  the  equation  of  the 
straight  line  ;  it  is  known  as  the  symmetrical  or  the  inter- 
cept form. 

Equation  [10]  may  also  be  derived  independently  of  equa- 
tion [9]  thus :  let  the  line  MN  (Fig.  42),  whose  equation 
is  to  be  found,  cut  the  axes  at  the  points  A  =  (^a,  0)  and 
^  =  (0,  5),  and  let  P  =  (x,y)  be  any  other  point  on  this 
line.     Connect  0  and  P;  then 

area  OPB  -h  area  OAP  =  area  OAB ; 

that  is,  ^hx  ■\-  \  ay  =  \ ah., 

X        V 

and,  dividing  by  \  ab,  this  equation  becomes  -  +  ^  =  1^  as 
above. 

EXERCISES 

1.   Show  that  equation  [10]  is  not  satisfied  by  the  coordinates  of  any 
point  except  those  lying  on  MN. 


84  ANALYTIC  GEOMETRY  [Ch.  V. 

2.  Write  down  the  equations  of  the  lines  through  the  following 
pairs  of  points : 

(a)     (3,  4)  and  (5,  2);  (y)     ("6,  1)  and  (-2, -5) ; 

(/?)     (3,  4)  and  (5,  "2);  (8)     (-15,-3)  and  (|,  ^)- 

3.  Write  the  equations  of  the  lines  which  make  the  following  inter- 
cepts on  the  X'  and  y-axes  respectively. 

(a)   4  and  7;      i/3)    "3  and  5;      (y)    |  and   -^;     (8)    -^  and  3  a. 

4.  What  do  equations  [9]  and  [10]  become  if  one  of  the  given 
points  is  the  origin? 

5.  By  drawing,  in  Fig.  42,  a  perpendicular^^il/  from  P  to  the  x-axis, 
derive  equation  [10]  from  the  similar  trisLngl^MpfiP  and  OAB. 

6.  Is  equation  [10]  true  if  P  is  on  MN  but  |/ot  between  A  and  B'^ 

7.  Are  equations  [9]  and  [10]  true  if  the  coordinate  axes  are  not 
at  right  angles  to  each  other? 

8.  Is  the  point  (3,  4^)  on  the  line  through  the  points  (2,  3)  and 
(5,  7)  ?  On  which  side  of  this  line  is  it  ?  AVhich  is  the  negative  side 
of  this  line?  ^ 

9.  What  intercepts  does  the  line  through  the  points  (1,  -6)  and 
(-3,  5)  make  on  the  axes  ? 

10.  The  vertices  of  a  triangle  are  :  (4,  -5),  (2,  3),  and  (3,  -6).  Find 
the  equations  of  the  sides;  also  of  the  three  medians;  then  find  the 
coordinates  of  the  point  of  intersection  of  two  of  these  medians,  and 
show  that  these  coordinates  satisfy  the  equation  of  the  other  median. 
What  proposition  of  plane  geometry  is  thus  proved? 

11.  Find  the  tangent  of  the  angle  (the  "  slope,"  cf .  Art.  27)  which 
the  line  in  exercise  9  makes  with  the  a;-axis.~ 

12.  Draw  the  line  whose  equation  is  -  +  ^  =  1,  and  then   find   the 

.J       o 

equations  of  the  two  lines  which  pass  through  the  origin  and  trisect  that 
portion  of  this  line  which  lies  in  the  first  quadrant. 

63.  Equation  of  straight  line  through  a  given  point  and 
in  a  given  direction  (cf.  Art.  44).  Let  P^  =  (ix^,  y^)  be 
the  given  point,  and  let  the  direction  of  the  line  be  given 
by  the  angle  XAP  =  6  which  the  line  makes  with  the 
a; -axis;  also  let  P={x,  y)  be  any  point  on  the  given  line 
and  denote  the  slope,  i.e.,  tan  ^,  bj  m.     Draw  the  ordinates 


52-53.  J 


THE  STRAIGHT  LINE 


85 


M^P^  and  MP,  and  through 
Pj  draw  P^R  parallel  to 
the  a;-axis  and  meeting  the 
ordinate  MP  in  R. 

Then,  in  triangle  RP^P, 
the  angle  RP^P  =  d ; 


Fig,  43. 


hence 


m 


tan  6  = 


RP 


P^R 


[11] 


X  —  x^ 
[Since  RP  =  y  —  y^  and  P^R  =  x  —  x^~\ ; 
that  is,  y -yi  =  in{oc  —  xi)f 

which  is  the  desired  equation. 

Cor.     If  the  given  point  be  ^=(0,  5),  ^.e.,  the  point  in 
which  the  line  meets  the  y-axis,  then  equation  [11]  becomes 

y  =  Qnx  +  b.        .  .  .  [12] 

Equation  [12]  is  usually  spoken  of  as  the  slope  form  of 
the  equation  of  the  straight  line. 


EXERCISES 

1.  What  do  the  constants  m  and  b  in  equation  [12]  mean?     Draw 

the  line  for  which  wi  =  4  and  6  =  3;  also  that  for  which  m  =  —  1  and 
7>  —  _  3 

2.  What  is  the  effect  on  the  line  [12]  of  a  change  in  b  while  m 
remains  the  same?     What  if  771  be  changed  and  b  left  unchanged? 

3.  Describe  the  effect  on  the  line  [11]  of  changing  ??i  while  x-^  and  y^ 
remain  the  same ;  also  the  effect  resulting  from  a  change  in  x^  while  m 
and  ?/^  remain  the  same. 

4.  Write  the  equation  of  a  line  through  the  point  (~3,  7),  and  mak- 
ing with  the  X-axis  an  angle  of  30'^;  of  -30°;  of   (=:^y'^  of  /^Zj"^^'"'* 

5.  Write  the  equations  of  the  following  lines  : 

(a)    slope  3,  y-intercept  8  ;     (/?)    slope  i,  3/-iutercept  "3 ; 
(y)     slope  -2,  ^/-intercept  ~|. 


86 


ANALYTIC  GEOMETRY 


[Ch.  V. 


6.  A  line  has  the  slope  6 ;  what  is  its  y-intercept  if  it  passes  through 
the  point  (7,  1)  ? 

7.  What  must  be  the  slope  of  a  line  whose  ?/-intercept  is  ~3,  in  order 
that  it  may  pass  through  the  point  (~5,  5)  ? 

8.  Is  the  point  (1,2)  01^  the  line  passing  through  the  point  (~2,  ~14), 
and  making  an  angle  tan~i^  with  the  a:-axis? 

9.  How  do  the  lines  y  =  ^x  —  1,  y  =  ^x-\-'l,  and  2?/  —  6a:  +  15  =  0 
differ  from  each  other?     AVhat  have  they  in  common?     Draw  these  lines. 

10.  What  is  common  to  the  lines  3/ =  3  a;  — *1,  2  3/ =  5  a;  —  2,  and 
7a;-32/  =  3? 

11.  What  is  the  slope  of  [9]  ?  of  [10]  ? 

12.  Derive  equation  [12]  independently  of  equation  [11]. 

54.  Equation  of  straight  line  in  terms  of  the  perpendicular 
from  the  origin  upon  it,  and  the  angle  which  that  perpendicular 
makes  with  the  i^-axis.     Let  HKhe  the  line  whose  equation 

K 

Y 


^^    a 


is  sought,  and  let  the  perpendicular  (^OIi=  p)  from  0  upon 
this  line,  and  the  angle  (a)  which  this  perpendicular  makes 
with  the  a;-axis,  be  given.  Also  let  P  =  (a:,  z/)  be  any  point 
on  ffK;    then  by  projection  upon  ON  (Art.  17), 

OM  cos  a  -\-  MP  sin  a  =  ON, 

^^e.,  xcosa  +  ysma=  p,      .         .         .       [13] 

which  is  the  required  equation. 

Equation  [13]  is  known  as  the  normal  form  of  the  equa- 
tion of  the  straight  line. 

In  the  following  pages  p  will  always  be  regarded  as  posi- 
tive, and  a  as  positive  and  less  than  860°. 


53-55.]  THE  STRAIGHT  LINE  87 

55.   Normal  form  of  equation  of  straight  line :  second  method. 

The  student  should  bear  in  mind  tliat  to  get  the  equation  of 
a  curve,  lie  has  merely  to  obtain  an  equation  that  is  satisfied 
by  the  coordinates  of  every  point  on  the  curve,  and  not 
satisfied  by  tlie  coordinates  of  any  other  point ;  and  that  it 
is  wholly  immaterial  what  particular  geometric  property  he 
may  employ  in  the  accomplishment  of  this  purpose.  This 
fact  is  already  illustrated  in  Art.  52,  where  equation  [10] 
was  obtained  in  two  ways,  while  Ex.  5,  p.  84,  gives  still  a 
third  method  by  which  the  same  equation  may  be  found. 
So  also  it  is  possible  to  derive  equation  [13]  by  other 
methods  than  that  employed  in  Art.  54.* 

U.g.,  in  Fig.  41  draw  a  perpendicular  from  0  to  the  line 
AB,  let  its  length  be  denoted  by  p,  and  let  a  be  the  angle 
which  it  makes  with  the  a:-axis,  then 

a  cos  a  =  p,  and  b  sin  a=  p, 

whence  a  =    ^    ,  and  b  =   .      . 

cos  a  sin  a 

Substituting  these  values  of  a  and  b  in  equation  [10],  it 

becomes     ^  y 

H — ^—  =  1,  i.e.,  X  cos  a  -{-  y  sin  a=  p, 


p  p 


cos  a      sin  a 
which  is  the  form  already  derived  in  Art.  54. 

Note.    In  Art.  2,  constants,  variables,  etc.,  were  illustrated  by  means 
of  a  triangle.     Now  that  the  student   has   learned  that   the  equation 

-  +  ^  =  1,  for  example,  represents  a  straight  line,  i.e.,  that  this  equation 
a      b 

is  satisfied  by  all  those  pairs  of  values  of  x  and  y  which  are  the  coordi- 
nates of  points  on  this  line,  a  somewhat  better  illustration  can  be  given. 
Both  X  and  y  are  variables,  but  are  not  independent ;  each  is  an  implicit 
function  of  the  other.  For  any  particular  line  a  and  h  are  constants,  but 
they  may  represent  other  constants  in  the  equation  of  another  line,  i.e., 
they  are  arbitrary  constants,  and  are  often  called  parameters  of  the  line. 

*  See  also  Ex.  6  below. 


88  ANALYTIC  GEOMETRY  [Ch.  V. 

EXERCISES 

1.  The  perpendicular  from  the  origin  upon  a  certain  line  is  5 ;  this 
perpendicular  makes  an  angle  of  -  with  the  a:-axis ;  what  is  the  equation 
of  the  line  ? 

2.  If  in  equation  [13]  p  is  increased  while  a  remains  the  same,  what 
is  the  effect  upon  the  line  ?  If  a  be  changed  while  p  remains  the  same, 
what  is  the  effect  ? 

3.  A  certain  line  is  3  units  distant  from  the  origin,  and  makes  an  angle 
of  120°  with  the  a;-axis ;  what  is  its  equation  ? 

4.  Given  a  =  30°,  what  must  be  the  length  of  p  in  order  that  the  line 
HK  (see  Fig.  44  a)  shall  pass  through  the  point  (7,  2)  ? 

5.  A  line  passes  through  the  point  (~3,  -4),  and  a  perpendicular  upon 
it  from  the  origin  makes  an  angle  of  45°  with  the  x-axis.  What  is  the 
equation  of  this  line  ? 

6.  In  Fig.  44  a  draw  through  M  a  line  parallel  to  HK,  meeting  ON  in 
/?;  then  draw  through  P  a  perpendicular  to  MR,  meeting  it  in  Q;  by 
means  of  the  figure  so  constructed  derive  equation  [13]  anew. 

56.  Summary.  The  results  of  Arts.  51-55  may  be  briefly 
summarized  thus : 

The  position  of  a  straight  line  is  determined  by  ;  (1)  two 
points  through  which  it  passes  ;  (2)  one  point  and  the  direc- 
tion in  which  the  line  passes  through  this  point.  Under  (1) 
there  is  the  special  case  in  which  the  two  given  points  are 
one  on  the  a;-axis  and  the  other  on  the  y-axis.  Under  (2) 
there  are  two  special  cases  :  (a)  when  the  given  point  is  on 
an  axis  (the  ?/-axis  say),  and  (/3)  when  the  point  is  given  by 
its  distance  and  direction  from  the  origin,  while  the  line 
whose  equation  is  sought  is  perpendicular  to  the  line  which 
connects  the  given  point  to  the  origin. 

Corresponding  to  these  two  general  and  three  special  cases, 
there  have  been  derived  five  standard  forms  of  the  equation 
of  the  straight  line,  viz.:  equations  [9],  [10],  [11],  [12], 
and  [13]. 

It  may  be  remarked  that  equations  [9]  and  [10]  are  inde- 
pendent of  the  angle  between  the  coordinate  axes,  while  [11], 


4 


55-57.]  THE  STRAIGHT  LINE  89 

[12],  and  [13]  (m,  a,  and^  retaining  their  present  meanings) 
are  true  only  when  the  axes  are  rectangular.  It  may  also  be 
pointed  out  that,  from  the  nature  of  its  derivation,  equa- 
tion [9]  is  inapplicable  when  the  line  is  parallel  to  either 
axis ;  equation  [10]  is  inapplicable  when  the  line  passes 
through  the  origin ;  and  equations  [11]  and  [12]  are  not 
applicable  when  the  line  is  parallel  to  the  ?/-axis. 

57.  Every  equation  of  the  first  degree  between  two  variables 
has  for  its  locus  a  straight  line.  It  will  probably  not  have 
escaped  the  reader's  notice  that  the  five  "  standard "  equa- 
tions (equations  [9]  to  [13])  of  the  straight  line,  which  have 
been  derived  in  Arts.  51  to  54,  are  each  of  the  first  degree. 
It  will  now  be  shown  that  every  equation  of  the  first  degree 
between  two  variables  has  a  straight  line  for  its  locus.  The 
most  general  equation  of  this  kind  may  be  written  in  the 

foi"m  Ax  +  By-\-C=0,     .         .         .       (1) 

where  A^  B^  and  C  are  constants,  and  neither  A  nor  B  is 
zero.* 

Let  Pi=(x^,  ^i),  P2  =  (^2'  ^2)'  aii^  ^3  =  (%  ^3)  be  any 
three  points  on  the  locus  of  equation  (1).  Draw  the  ordi- 
nates  M^P^,  M^P^^  and  M^P^\  also  draw  HP,^  and  KP^ 
parallel  to  the  a:-axis.  sJ^ 

Then,  by  Art.  35  (1),  /Ng 

A..3+%3  +  (7=0...(4)     L'^    I     I ^3^< 

/  Fig. 45. 

*  If  either  A  or  B,  say  A^  is  zero,  then  the  equation  may  be  written  in  the 

C 
form  :  y  = ,  which  is  the  equation  of  a  straight  line  parallel  to  the  x-axis, 

^             C 
and  at  the  distance from  it  [cf.  Art.  33,  (2)]. 


90  ANALYTIC  GEOMETRY  [Ch.  V. 

By  subtracting  eq.  (3)  from  eq.  (2),  and  also  eq.  (4)  from 
eq.  (3),  the  two  equations 

and  ^(^2 -^'3) +  ^(^2 -^3)=^' 

are  obtained.     These  give 

a^l2^-i  and  ^^^^3^-4;     ...      (5) 
hence,    VlZll  =  yJLlll,  ...  (6) 

2^1  —  ^2        ^^2         *^3 
But  t/i   -  ^2  =  ^Pv  ^1  -  ^2  =  -^1^2  =  -^^2^ 

y<2,-yz  =  KPv  ^^^^  ^2  -  ^3  =  -^2^3  =  --^A ' 

hence,  from  eq.  (6),        -=pi  =  j[p' 
Also,  by  construction, 

hence,     triangle  HP^P^  is  similar  to  triangle  KP^P^, 

and  Z.P^P^E  =  /.PJP^K; 

.  • .  Z  PiP2S^  +  Z^P^^  +  ZirP2P3 

=  Z  P^P^K  +  Z.  P^KP^  +Z  XP2P3  =  2  rt.  Zs; 

^.e.,  P2  lies  on  the  straight  line  joining  P^  and  P3.  But, 
since  P^  is  any  point  on  the  locus  of  Ax-\-  By  +  C  =  ^,  hence 
all  points  of  this  locus  lie  on  the  same  straight  line  P^P^, 
which,  therefore,  constitutes  the  locus  of  Ax  -\-  By  +  C  =^. 
Since  this  demonstration  does  not  depend  upon  the  angle 
ft),  therefore  it  applies  whether  the  axes  are  oblique  or  rec- 
tangular ;  hence  the  theorem  :  every  equation  of  the  first 
degree-  between  two  variables,  ivJien  interpreted  in  Cartesian 
coordinates,  represents  a  straight  line.* 

*  This  conclusion  may  also  be  drawn  thus :    clear  equation  (6)  of  frac- 
tions, transpose  all  the  terms  to  the  first  member,  and  multiply  by  ^  sin  w ; 


67-58.]  THE   STRAIGHT  LINE  91 

Because  of  this  fact,  such  an  equation  is  often  spoken  of 

as  a  linear  equation. 

N'oTE.  In  the  equation  Ax  -\-  By  +  C  =  0,  there  are  apparently  three 
constants;  in  reality,  there  are  but  two  independent  constants,  viz.  the 
ratios  of  the  coefficients  (cf.  Art.  88).  This  corresponds  to  the  fact  that 
a  straight  line  is  determined  geometrically  by  two  conditions. 

58.  Reduction  of  the  general  equation  Aoc  +  By  +C  =  0  to 
the  standard  forms.  Determination  of  «,  b,  m,  i>,  and  a  in 
terms  of  A,  B,  and  O.* 

(1)  Reduction  to  the  standard  form  -  +  ^  =  1  (^symmetric 
or  intercept  form). 

That  the  equation 

Ax  +  By-\-C  =  0       .         .         .      (1) 

represents  some  straight  line  lias  just  been  shown  (Art.  57) ; 
again,  since  multiplication  by  a  constant,  and  transposition, 
do  not  change  the  locus  (Art.  38),  therefore 

""    +-^=lt       .         .         .       (2) 


A         B 

represents  the  same  line.  But  equation  (2)  is  in  the  re- 
quired form  (Art.  52),  and  its  intercepts  are  : 

a  =  — 7,  and  6=—  —  . 
A  B 

(2)  Reduction  to  the  standard  form  y  =  mx  +  h  Qslope 
form). 

the  resulting  equation  asserts  [see  Art.  29,  (1)]  that  the  area  of  the  triangle 
formed  by  the  points  Pi,  Pj,  and  P3,  is  zero;  i.e.,  these  three  points  lie  on  a 
straight  line  ;  but  they  are  any  three  points  on  the  locus  of  Ax  +  By-{-  C  =0, 
hence  that  locus  is  a  straight  line. 

*  These  reductions  constitute  a  second  proof  of  the  theorem  of  Art.  57. 

t  If  O  =  0,  the  line  represented  by  (1)  goes  through  the  origin,  and  the 
symmetric  form  of  the  equation  is  inapplicable  (Art.  56) ;  but,  in  that  case, 
the  above  reduction  also  fails,  since  it  is  not  permissible  to  divide  the  n^pn- 
bers  of  an  equation  by  zero. 


92  ANALYTIC   GEOMETRY  [Ch.  V. 

The  equation  Ax  +  By  +  (7=0  has  the  same  locus  as.  has 
the  equation 

(see  Art.  38);   but  this  is  the  equation  (Art.  53)  of  a  line 
drawn  through  the  point  fO,  —  ^),   and  making  with  the 

2:-axis  the  angle  0  =  tan  ~  M  —  ^ )  5   hence  equation  (3)  is  in 
the  required  form,  and 

m  =  — — ,  and   6  =  —  —  . 

(3)    Reduction  to  the  standard  form   xcos  a  -\-  y  sin  a  =  p 
(normal  form). 

If  equation  (1)  and 

X  cos  a  -{-  y  sin  a  =  p        .  .  .  (4) 

represent  the  same  line,  then  they  differ  merely  by  some 
constant  multiplier,  say  k  (cf.  Art.  38).     Then 

kAx  +  kBy  +  kC  =  x  cos  cc  +  y  sin  a  ~  p  =  0  ; 
.  • .  kA  =  cos  a,  kB  =  sin  a,  and  kC  =  —p ; 
. • .  k'^A^  +  k^B^  =  cos ^a  +  sin ^a  =  l; 

whence  k  =  —      "         ; 

hence  cos  a  =  —  .  sm  a  = 


and  p 


V^2^^2  -VA^  +  B^ 

O 


*  If  5  =  0,  the  line  represented  by  equation  (1)  is  parallel  to  the  y-axis, 
ai*  the  slope  form  of  the  equation  is  inapplicable  (Art.  56) ;  but,  in  that  case, 
the  above  reduction  also  fails. 


58.]  THE  STRAIGHT  LINE  93 


wherein  the  algebraic  sign  of  VjL^  +  B^  is  to  be  chosen  so 
as  to  make  — — z^^^^  positive,  since  p  is  to  be  always  posi- 

tive  (Art.  54);  ^.e.,  the  sign  of  VJ.^  +  B^  is  to  be  opposite  to 
that  of  the  number  represented  by  C. 

Hence,  to  reduce  equation  (1)  to  the  normal  form,  ^.e.,  to 
the  form  of  equation  (4),  it  is  only  necessary  to  divide  equa- 
tion (1)  by  V^2  +  B^,  with  the  sign  properly  chosen,  and 
transpose  the  constant  term  to  the  second  member.     This 

gives 

A  ,         B  -C 


V J.2  +  B^  VW+W  V^2  4_  ^2 

(4)  Another  method  for  reduction  to  the  normal  form. 

If  the  equation  Ax  +  By  +  C  =  0  and  x  cos  a  +  y  sin  a=  p 
represent  the  same  line,  then  they  must  have  the  same 
^-intercept  and  the  same  slope,  i.e,^ 

(5) 

and  —  ^  = ; •  •  •  C'^) 

B  sm  a 

Squaring  eq.  (6),  and  adding  1  to  each  member,  gives 

A^  +  B^  _  cos^  a  +  sin^  a 
B^  sin^  a 


0 
B 

P 
sin  a 

A 

cos  a 

B 

sin  a 

sma  = 


sm-^a 
B 


A  —  C 

whence  cos  a  =     ^  and    p  =  —  . 


94  ANALYTIC  GEOMETRY  [Ch.  V. 

as  before.     These,  then,  are  the  values  of  p,  sin  a,  and  cos  a, 
which  are  to  be  substituted  in  x  cos  a-\-  y  sin  a  =p. 

Hence  ^        r.  ^    .    ^      -v—         ^ 


V^2  ^  ^2  V^2  +  ^2  V  J12  +  J52' 

is  an  equation  representing  the  same  locus  as  Ax  +  By  +  (7=  0, 
and  having  the  normal  form. 

59.  To  trace  the  locus  of  an  equation  of  the  first  degree.     In 
Art.  57  it  was  proved  that  the  locus  of  an  equation  of  the 

first  degree  in  two  variables  is  a 
straight  line ;  but  a  straight  line 
is  fully  determined  by  any  two 
points  on  it ;  hence,  to  trace  the 
locus  of  a  first  degree  equation  it 
is  only  necessary  to  determine  two 
of  its  points,  and  then  to  draw  the 
indefinite  straight  line  through  them.  The  two  points  most 
easily  determined,  and  plotted,  are  those  in  which  the  locus 
cuts  the  axes  ;  they  are  therefore  the  most  advantageous 
points  to  employ.  If  the  line  is  parallel  to  an  axis,  then 
only  one  point  is  needed. 

E.g.^  to  trace  the  locus  of  the  equation 

the  ordinate  of  the  point  in  which  this  line  crosses  the  rr-axis 
is  0  ;  let  its  abscissa  be  x-^,  then  (x-^^  0)  must  satisfy  the  equa- 
tion  2a;-32/  +  12  =  0; 

hence  2:^1-3.0  +  12  =  0, 

whence  ar^  =  —  6, 

i.e.^  the  line  crosses  the  a;-axis  at  the  point  (~6,  0).  In  like 
manner  it  is  shown  that  it  crosses  the  y-axis  at  the  point 
(0,  4).     Therefore  LM  is  the  locus  of  2  2:  -  3  ?/  +  12  =  0. 


58-60.]  THE  STRAIGHT  LINE  95 

60.   Special   cases   of    the    equation    of    the   straight    line 

Aoc  +  By  -\-  C  =  O.     This  equation,  written  in  tlie  intercept 
form  [Art.  58  (1)]  becomes 

X  y 


C  '^      n~   ''      *        *        *        ^  -^ 
~A      ~B 

If  in  equation  (1),  A  is  made  smaller  and  smaller  in  com- 
parison with  (7,  then  the  a^-intercept  [ — -]  becomes  larger 

and  larger ;  if  ^  =  0  in  comparison  with  (7,  the  2:-intercept 
grows  infinitely  large,  the  line  (1)  becomes'  parallel  to  the 
a;-axis,  and  its  equation  becomes 

^   ^     y       -x  .  :  .    ..         ^ 


— +  -^  =  1;  i.e.,  y  =  - 
B 


C""'  ....,^---g, 


which  agrees  with  the  foot-note  of  Art.  57. 

Similarly,  if  ^  =  0  in  comparison  with  C^  the  line  (1)  be- 
comes parallel  to  the  ?/-axis,  and  its  equation  becomes 

0 
.  =  --. 

If  both  A  and  B  approach  zero  simultaneously  in  compari- 
son with  C,  then  both  the  intercepts  become  indefinitely 
large,  and  the  line  (1)  recedes  farther  and  farther  from  the 
origin. 

In  accordance  with  what  has  just  been  said,  a  line  that  is 
wholly  at  infinity  might  have  its  equation  written  in  the 

form  0 .  a:  4-  0 .  ^  +  (7  =  0,       .       .        .         (2) 

or,  as  it  is  sometimes  written,        (7=0;       .       .       .         (3) 

but  equations  (2)  and  (3)  are  merely  abbreviations  for  the 
statement :  "  As  both  A  and  B  approach  zero  in  comparison 
with  (7,  the  line  moves  farther  and  farther  from  the  origin." 


96  ANALYTIC  GEOMETRY  [Ch.  Y. 

EXERCISES 

1.  Reduce  the  following  equations  to  the  intercept  (symmetric) 
form,  and  draw  the  lines  which  they  represent : 

(a)     3x-2z/  +  12  =  0;  (^8)    3x  -  2  z/ +  1  =  5a;  +  3; 

(y)    2y  =  lb-y+^X',  .        (8)     ^Jzl^jJ.  =  9. 

^+  i  y 

2.  Reduce  to  the  slope  form,  and  then  trace  the  loci : 

(a)     7a:-53/  + 6(?/- 3a:)=- 10a;4-4;         (yg)    3a: +  2?/ +  6  =  0; 

(y)    3a:+5  =  3-2^. 

Which  is  the  positive  side  of  the  line  (^)  ?     (cf.  foot-note,  Art.  43.) 

3.  Reduce  to  the  normal  form,  and  then  trace  the  loci : 
(a)     3a: +  4?/ =  15;  (yg)    3a;  -  4?/ +  15  =  0  ; 
(y)    a;  -  3  ?/  =  5  +  6 a; ;                   (8)     ^x  =  y  -b. 

4.  Show  that  the  lines  3  a:  +  5  =  ?/  and  6  a:  —  2  ?/  =  81  are  parallel. 

5.  What  is  the  slope  of  the  line  between  the  two  points  (3,  -1)  and 
(2,  2)  ?  What  is  its  distance  from  the  origin  ?  Which  is  its  negative 
side? 

6.  A  line  passes  through  the  point  (5,  6)  and  has  its  intercepts  on 
the  axes  equal  and  both  positive.  Find  its  equation  and  its  distance 
from  the  origin. 

7.  A  straight  line  passes  through  the  point  (1,  -2)  and  is  such  that 
the  portion  of  it  between  the  axes  is  bisected  by  that  point.  What  is  the 
slope  of  the  line? 

8.  What  are  the  intercepts  which  the  line  through  the  points  (-1,  3) 
and  (6,  7)  makes  on  the  axes  ?     Through  the  points  («,  2  a)  and  {h,  2h)l 

9.  What  system  of  lines  obtained  by  varying  the  parameter  h  is  rep- 
resented by  the  equation  y  =  6  a:  +  &  ? 

10.  What  system  of  lines  obtained  by  varying  the  parameter  m  is 
represented  by  the  equation  y  =  mx  +  6  ? 

11.  What  family  (system)  of  lines  obtained  by  varying  the  parameter 
a  is  represented  by  the  equation  x  cos  a  +  y  sin  a  =  5  ?  To  what  curve  is 
each  line  of  the  family  tangent  ? 

12.  Find  cos  a  and  sin  a  for  the  lines 

(a)     y  =  mx  +  h,  (/S)      -  +  {=1, 

a      0 

(y)     ?  =  ?,  (8)     7a:-5^+l=0. 


60-61.] 


THE  STRAIGHT  LINE 


97 


13.  Find  by  means  of  cos  a  and  sin  a  what  quadrant  is  crossed  by 
each  of  the  lines  : 

(a)     3x  +  2  =  2y;       (^)     5a:  +  3y  +  15  =  0;       (y)     x-y/^y-\0  =  0. 

14.  What  must  be  the  slope  of  the  line  '^x  -  ky  =  11  in  order  that  it 
shall  pass  through  the  point  (1,  3)?  Can  k  be  determined  so  that  the 
line  will  pass  through  the  origin? 

15.  Determine  the  values  oi  A,  B,  C  in  order  that  the  line 

Ax -{- By -\- C  =  0 
shaU  pass  through  the  points  (3,  0)  and  (0,  -12).     [Art.  57,  Note.] 

16.  Derive  equation  [9]  by  supposing  (x^,  y{)  and  (a-'g,  2/2)  ^^  ^®  ^^^ 
points  on  the  line  y  =  mx  +  &-;  and  thence  finding  values  for  m  and  h. 

17.  Find  the  slopes  of  the  lines  2 ?/  —  3 a;  =  7  and  32/  +  2x  —  11  =  0; 
and  thence  show  that  these  lines  are  perpendicular  to  each  other. 

18.  Find  cos  a  for  each  of  the  lines  7a;  +  ?/  —  9  =  0  and  x  —  7y  +  2  =  0, 
and  then  show  that  the  two  lines  are  perpendicular  to  each  other. 

'     19.   Show  by  means  of:  (1)  the  slopes;  (2)  the  angles;  that  the  lines 

2y-3x=:7,        2y-3a;  +  5  =  0,         10y-15x  +  c  =  0 
are  all  parallel. 

20.  Reduce  the  equation  Ax  -{-  By  +C  =0  to  the  normal  form, 
i.e.,  to  the  form  x  cos  a  +  y  sin  a=p.  Suggestion  :  the  two  equations  as 
representing  the  same  line,  make  the  same  intercepts  on  the  axes. 


61.   To  find  the  angle  made  by  one  straight  line  with  another. 
Let  the  equations  of  the  lines  be 

i/  =  mjX-\-hi,  ,  .  .   (1) 

and    ^  =  vYiiX  +  52,...   (2) 

where  mi  =  tan  By-,  ^2  =  tan  ^2? 
and  ^1,  ^2  are  the  angles  which 
these  lines  make,  respec- 
tively, with  the  a^-axis.  It  is 
required  to  find  the  angle  0, 
measured  from  line  (2)  to  line  (1). 

TAN.  AN.  GEOM.  —  7 


\  X 


98  ANALYTIC  GEOMETRY  [Ch.  V. 

Since  <^  =  ^i  —  O2', 

,        ,         tan  6x  —  tan  62  ^  a   j.    -1  ^\ 

tan  6=- i-- ^,       (Art.  16) 

^      1  +  tan  (9i  •  tan  62  ^ 

'  1  +  niiin2  L     J 

If  the  angle  were  measured  from  line  (1)  to  line  (2)  it 
would  be  the  negative,  or  else  the  supplement,  of  0  ;  in 
either  case  its  tangent  would  be  the  negative  of  that  given 
by  formula  [14]. 

If  the  equations  of  the  lines  had  been  given  in  the  form  : 

A^  +  ^i^  +  <^i  =  0,       ...       (3) 
and  A2X  +  Boi/  -f-  (^2  =  0,       .        .        .       (4) 

A  A 

then  mi=  — — \  7712  =  —  ^^  and  formula  [14]  becomes 

tan0  =  lA^^.4,^L:^A^.     .     .     [15] 
1^4i42       A,A2  +  B,B2  ^     -' 

B^B2 

EXERCISES 

Find  the  tangent  of  the  angle  from  the  first  line  to  the  second  in  each 
of  the  following  cases,  and  draw  the  figures : 

1.  3a; -43/ -7  =  0,  2a:-?/ -3  =  0; 

2.  5a;  +  12  3/  +  l=0,  a;-2?/  +  6  =  0; 

3.  2a:  =  3?/-|-9,         6?/  =  4a;+2; 

a      b  ah 

5.   a;  cos  a  -\-  y  sin  a  =  p,        -  +  ^  =  1. 

a      6 

62.   Condition  that  two  lines  are  parallel  or  perpendicular. 

From  formula   [14]   can  be  seen  at  once  the  relations  that       ; 


61-62.]  THE  STRAIGHT  LINE  99 

must  hold  between  rrii  and  mg  if  tlie  lines  (1)  and  (2) 
(Art.  61)  are  parallel  or  perpendicular.  If  these  lines  are 
parallel,  then  <^  =  0,  and  therefore  tan  0  =  0; 

hence  —^ =  0, 

1  +  ^17/12 

I.e.,  nii=in29 

which  is  the  condition  that  lines  (1)  and  (2)  are  parallel.* 
This  condition  is  also  evident  from  a  mere  inspection  of 
equations  (1)  and  (2). 

If  the  lines  (1)  and  (2)  (Art.  61)  are  perpendicular,  then 
(j)  =  90°  and  tan  (f)  =  cc  , 

^.e.,  ; — —  =  GO  ,  hence  1  +  wiimg  =  0, 

1  +  mim2 

1 

^.e.,  fn2  = 9 

mi 

which  is  the  condition  that  (1)  and  (2)  are  perpendicular. 
So  also  from  [15]  the  lines 

Aix  +  Bii/  +  (7^  =  0    and    A2X  +  A«/  +  C2  =  0 

are  parallel  if  (and  only  if)  ^2^1  —  ^1^2  =  0, 

^'.e.,  if  Ai  :^i  =  J.2:  ^2; 

and  they  are  perpendicular  if  (and  only  if)  J.i^2  +  -^i^2=0? 

i.e.,  if  Ai'.  Bi=—B2:  A- 

The  condition  just  found  enables  one  to  write  down  readily 
the  equations  of  lines  that  are  parallel  or  perpendicular  to 
given    lines,    and  which  also    pass    through    given    points. 

*  It  must  not  be  forgotten  that  this  conclusion  is  drawn  only  for  lines 
that  are  not  perpendicular  to  the  a:-axis ;  because  if  the  lines  are  perpen- 
dicular to  the  X-axis  then  equations  (1)  and  (2)  are  inapplicable  (cf.  Art.  56). 


100  ANALYTIC  GEOMETRY  [Ch.  V. 

E.g.^  let  it  be  required  to  write  the  equation  of  a  line  that  is 

parallel  to  the  line 

y  =  ^x+'J.  .  .  .  (1) 

The  slope  of  this  line  is  3,  hence  any  other  line  whose  slope 
is  3  is  parallel  to  the  given  line, 

i.e.,  7/  =  Sx-\-h,         .  .-         .  (2) 

is,  for  all  values  of  5,  parallel  to  line  (1). 

If  it  be  required  that  the  line  (2)  shall  also  pass  through 
a  given  point,  (1,  5)  for  example,  it  is  only  necessary  to 
determine  rightly  the  value  of  b.  This  is  done  by  remem- 
bering that  if  the  line  (2)  passes  through  the  point  (1,  5), 
then  these  coordinates  must  satisfy  equation  (2), 

i.e.,  5  =  3  •  1  4- J,  whence  5  =  2. 

Therefore  the  line  i/  =  Sx  +  2  is  not  only  parallel  to  the 
line  ?/  =  3  a;  +  7,  but  also  passes  through  the  point  (1,  5). 

Similarly  y  =  —  i  a;  +  6,  whatever  the  value  of  5,  is  per- 
pendicular to  y  =  Sx-\-  1. 

Again,  the  line  Sx  +  5i/-{-k  =  0,  whatever  the  value  of  k, 
is  parallel  to  the  line  3a:  +  5y  — 15  =  0;  and  the  line 
5x  —  Sy'{-k  =  0  is  perpendicular  to  3  a;  -1-  5  y  —  15  =  0. 
Here  again  the  arbitrary  constant  k  may  be  so  determined 
that  this  line  shall  pass  through  any  given  point.  So  also 
the  lines  A^x  +  B^y  +  (7^  =  0  and  A^x  +  B^y  -{-  C^^  —  0  are 
parallel,  while  A^x  -|-  B^y  +  6^  =  0  and  B^x  —  A-^y  -\-  C^  =  0 
are  perpendicular  to  each  other. 

This  condition  for  parallelism  and  for  perpendicularity 
of  two  lines  may  also  be  stated  thus  :  two  lines  are  parallel 
if  their  equations  differ  (or  may  be  made  to  differ)  only  in 
their  constant  terms ;  two  lines  are  'perpendicular  if  the  coeffi- 
cients of  X  and  y  in  the  one  are  equal  (or  can  be  made  equal), 
respectively.,  to  the  coefficients  of  —y  and  x  in  the  other. 


62-63.]  THE  STRAIGHT  LINE  101 

EXERCISES 

1.  Write  down  the  equations  oi"  the  set  of  lines  parallel  to  the  lines : 

(a)     y  =  Qx-2;       (f3)      3x-7y  =  3; 

(y)     a:cos30°  +  2/sin30°  =  8;       (8)     ^-|  =  1. 

2.  Explain  why  it  is  that  the  constant  term  in  the  answers  to  Ex.  1 
is  left  undetermined  or  arbitrary. 

3.  Find  the  tangent  of  the  angle  between  the  lines  (a)  and  ((3)  in 
Ex.  1 ;  also  for  the  lines  (/3)  and  (8),  and  (a)  and  (8)  of  Ex.  1. 

4.  Write  the  equations  of  lines  perpendicular  to  those  given  in  Ex.  1. 

5.  By  the  method  of  Art.  62  find  the  equation  of  the  line  that  passes 
through  the  point  ("9,  1),  and  is  parallel  to  the  line  y  =  Qx  —  2. 

6.  Solve  Ex.  4  by  means  of  equation  [11],  Art.  53. 

7.  Find  the  equation  of  the  line  that  is  parallel  to  the  line  Ax  +  By 
+  C  =  0  and  that  passes  through  the  point  (x^,  y^  ;  make  two  solu- 
tions, one  by  the  method  of  Ex.  4,  and  the  other  by  Ex.  5. 

Find  the  equation  of  the  straight  line 

8.  through  the  point  (2,  —5)  and  parallel  to  the  line  y  =  2  x  +  1 . 

9.  through  the  point  ("l,  —1)  and  perpendicular  to  y  =  2  x  -\-  7  ] 
solve  by  two  methods. 

10.  through  the  point  (0,  0)  and  parallel  to  the  line 

3        7         X  -  y  -^  1 

r-iy= — 9 — 

11.  perpendicular  to  the  line  2y-\-1x  —  l  =  Q,  and  passing  through 
the  point  midway  between  the  two  points  in  which  this  line  meets  the 
coordinate  axes. 

12.  Find  the  foot  of  the  perpendicular  from  the  origin  to  the  line 
5x-7?/  =  2. 

63.   Line  which  makes  a  given  angle  with  a  given   line. 

The  formula 

,         tan  6.  —  tan  0^      ^  k    ,    n-i^ 

tan  6  =  3 ,  ^  ^    1-    (Art.  61) 

^      1  4-  tan  6^  tan  0^    ^  ^ 

states  the  relation  existing   between   the   tangents   of   the 
angles  6^^  0^,  and  cf)  (see  Fig.  47)  ;  hence  if  any  two  of  these 


102 


ANALYTIC  GEOMETRY 


[Ch.  V. 


angles  are  known,  this  equation  determines  the  value  of  the 
third.  Thus  this  formula  may  be  employed  to  determine 
the  slope  of  a  line  that  shall  make  a  given  angle  with  a 
given  line. 

E.g.^  given  the  line  3?/  —  5a;  +  7  =  0,  to  find  the  equation 
of  a  line  that  shall  make  an  angle  of  60°  with  this  line. 
Here  (j)  =  60°,  i.e.,  tan  </>  =  V3,  and  if  6^  be  the  angle  which 
the  given  line  makes  with  the  2:-axis,  and  6^  that  made  by 
the  line  whose  equation  is  sought,  then  tan  ^^  =  |-.  Substi- 
tuting these  values  in  the  above  formula,  it  becomes 


whence 


V3^ 


I  —  tan  ^2 
1  4- 1  tan  ^2' 


,      .       5-3V3         ,  5-3V3 

tan  To  = — =,  and  y  = z — —  'X  -\-  k 

^     3  +  5V3  ^      3  +  5V3 


is  the  equation  of  a  line  fulfilling  the  required  conditions,  — 
k  may  be  so  determined  that  this  line  shall  also  pass 
through  any  given  point. 

It  is  to  be  remarked  that  through  any  given  point  there 
may  be  drawn  two  lines,  each  of  which  shall  make,  with  a 
given  line,  an  angle  of  any  desired  magnitude. 


E.g.,  through  P^=  {x^,  y^  the  lines  (1)  and  (2)  may  be 
so  drawn  that  each  shall  make  an  angle  </>  with  the  given 


63.]  THE  STRAIGHT  LINE  103 

line  LM.     Let  line  (1)  make  an  angle  0^^  line  (2)  an  angle 
^2?  and  LM  an  angle  ^3,  with  the  a;-axis  ;  then 

<^  =  6>i  -  ^3,  and  180-  </>  =  ^2  -  ^3 » 
which  gives 

tan  ^1  —  tan  ^q  ,        ,        ,        tan^,  — tan^o 

tan  6  =  ^ r-^— 7^ ^,  and   —  tan  6  =  z ^      n  ^ — n 

^      1  +  tan  ^1  tan  0^'  ^      1  -f  tan  6^  tan  d^ 

In  these  equations  cf)  and  0^  are  known,  hence  tan  0^  and 
tan ^2  ^^^  ^®  found.  Having  found  tan^^  and  tan  ^2  ^^^ 
equations  of  lines  (1)  and  (2)  may  at  once  be  written  down, 
either  by  means  of  equation  [11],  or  by  the  method  employed 
in  Art.  62. 

EXERCISES 

1.  Find  the  equations  of  the  two  Hnes  which  pass  through  the  point 
(5,  8),  and  each  of  which  makes  an  angle  of  45°  with  the  Ime  2  x  —  3y  =  Q. 

2.  Show  that  the  equations  of  the  two  straight  Unes  passing  through 
the  point  (3,  -2)  and  inclined  at  60°  to  the  line  x  V3  +  ?/  =  1  are 

y  +  2  =0  and  ?/-xV8  +  2  +  3V3  =  0. 
Find  the  equation  of  the  straight  line 

3.  making  an  angle  of  —  —  with  the  line  3  a:  —  4  y  =  7 ;  construct  the 

figure.     Why  is  there  an  undetermined  constant  in  the  resulting  equation  ? 

4.  making  an  angle  of  +  60°  with  the  line  5x+12y-\-l  =  0;  con- 
struct the  figure. 

5.  making  an  angle  of  —  30°  with  the  line  x  —  2y  +  l=0,  and 
passing  through  the  point  (1,3);  making  an  angle  of  +  30°,  and  passing- 
through  the  same  point. 

6.  making  an  angle  of  it  135°  with  the  line  x  +  y  =  2,  and  passing 

through  the  origin. 

J)  or      7J 

7.  making     the    angle    tan"^  -  with  the  line  -  +  ^  =  1,  and  passing 

f      h\  ^  a     b 

through  the  point  f  -,  -  j . 

8.  Find  the  equation  of  a  line  through  the  point  (4,  5)  forming  with 
the  lines  2x  —  y-\-^  =  0  and  3y  +  6a;  =  7a  right-angled  triangle.  Find 
the  vertices  of  the  triangle  (two  solutions). 


104  ANALYTIC  GEOMETRY  [Ch.  V. 

9.   Show  that  the  triangle  whose  vertices  are  the  points  (2, 1),  (3,  -2), 
(-4,  -1)  is  a  right  triangle. 

10.  Prove  analytically  that  the  perpendiculars  erected  at  the  middle 
points  of  the  sides  of  the  triangle,  the  equations  of  whose  sides  are 

X  +  y  -\-l  =  0,  ?>  X  +  b  y  +  11  =  0,  and  x-\-2y  +  ^  =  0, 

meet  in  a  point  which  is  equidistant  from  the  vertices. 

11.  Find  the  equations  of  the  lines  through  the  vertices  and  perpen- 
dicular to  the  opposite  sides  of  the  triangle  in  exercise  10.  Prove  that 
these  lines  also  meet  in  a  common  point. 

12.  A  line  passes  through  the  point  (2,  -3)  and  is  parallel  to  the 
line  through  the  two  points  (4,  7)  and  (-1,-9)  ;  find  its  equation. 

13.  Find  the  equation  of  the  line  which  passes  through  the  point  of 
intersection  of  the  two  lines  10a:  +  5?/  +  ll  =  0,  and  a;  +  2?/  +  14  =  0, 
and  which  is  perpendicular  to  the  line  x-f7?/+l  =  0. 

This  problem  may  be  solved  by  first  finding  the  point  of  intersection 
i^i  —  ^)  of  the  two  given  lines,  and  then,  by  formula  [11]  (see  also 
Art.  62),  writing  the  equation  of  the  required  line,  viz. : 

2/  +  ^  =  7  (:r  -  ^), 
which  reduces  to  7  a:  —  ?/  =  31. 

The  problem  may  also  be  solved  somewhat  more  briefly,  and  much 
more  elegantly,  by  employing  the  theorem  of  Art,  41.  By  this  theorem 
the  equation  of  the  required  line  is  of  the  form 

10  a:  +  5  ?/  +  11  +  A'  (a:  +  2  ?/  +  14)  =  0, 
i.e.,  (10  +  ;^)  a:  +  (5  +  2  yt)  !/  +  11  +  14  ^  =  0. 

It  only  remains  to  determine  the  constant  h,  so  that  this  line  shall 
be   perpendicular  to   a:  +  7  ?/  +  1  =  0.      By  Art.   62   its   slope  must   be 

=  7,     hence — —  =  7,     whence  ^  =  —  3. 

Substituting  this  value  of  Tc  above,  the  required  equation  becomes 
7  a:  —  2/  =  31,  as  before. 

14.  By  the  second  method  of  exercise  13  find  the  equation  of  the  line 
which  passes  through  the  point  of  intersection  of  the  two  lines  2  a:  +  y  =  5 
and  a;  =  3y  —  8,  and  which  is :  (1)  parallel  to  the  line  4?/  =  6  a:  +  1 ; 
(2)  perpendicular  to  this  line ;  (3)  inclined  at  an  angle  of  60°  to  this 
line ;   (4)  passes  through  the  point  ("1,  3). 

15.  Solve  exercise  10  by  the  method  of  exercise  14. 


63-64.]  THE  STBAIGHT  LINE  105 

16.  Do  the  lines  2  a;  +  3  y  =  13,  5x  -  y  =  7,  and  a:-4?/  +  10  =  0 
meet  in  a  common  point  ?  What  are  the  angles  they  make  with  each 
other  ? 

17.  Find  the  angles  of  the  triangle  of  exercise  10. 

18.  When  are  the  lines 

x-{-(a-{-h)y  +  c  =  0  and  a(x  +  ay)  -^  h  (x  —  by)  +  d  ■=  0 
parallel?  when  perpendicular? 

19.  Find  the  value  oi  p  for  each  of  the  two  parallel  lines 

y  =  3  X  -\-  7  and  y  =  S  x  —  5] 
and  hence  find  the  distance  between  these  lines  [cf.  Art.  58  (3)  and  (4)]. 

20.  What  is  the  distance  between  the  two  parallel  lines 

5x  -3y  +  6  =0  and  Qy  -  10a;  =  7? 

21.  Find  the  cosine  of  the  angle  between  the  lines 

y  -  4:x  +  8  =  0  and  y-6x  +  9  =  0. 

22.  What  relation  exists  between  the  two  lines 

y  =  3x  +  7  and  y  =  -  3 a:  -  3 ? 

23.  Find  the  angle  between  the  two  straight  lines  3  a:  =  4  ?/  +  7  and 
5y  =  12a;  +  6;  and  also  the  equations  of  the  two  straight  lines  which 
pass  through  the  point  (4,  5)  and  make  equal  angles  with  the  two  given 
lines. 

24.  Find  the  angle  between  the  two  lines 

3x  +  y  +  12  =  0  and  x  +  2y-l  =  0. 

Find  also  the  coordinates  of  their  point  of  intersection,  and  the  equations 
of  the  lines  drawn  perpendicular  to  them  from  the  point  (3,  ~2). 

64.  The  distance  of  a  given  point  from  a  given  line.  This 
problem  is  easily  solved  for  any  particular  case  thus  :  find 
the  equation  of  the  line  which  passes  through  the  given 
point  and  which  is  parallel  to  the  given  line  (Art.  62),  then 
find  the  distance  (jt?)  from  the  origin  to  each  of  these  two 
lines  [Art.  58,  (3)  and  (4)],  and  finally  subtract  one  of  these 
distances  from  the  other ;  the  result  is  the  distance  between 
the  given  line  and  the  given  point. 


106 


ANALYTIC  GEOMETRY 


[Ch.  V. 


Fig.  49. 


E.g.^  find  the  distance  of  the  point  Pi=(2,  |)  from  the 

^^^^  3a;4-i?/-7  =  0.       .        .        .        (1) 

Let  line  (1)  be  the  locns 
of  equation  (1),  andPj  be 
the  given  point.    Through 
Pj  draw  the  line  (2)  par- 
allel to  line  (1),  also  draw 
QP^  perpendicular  to  line 
(1),    OE^(  =  p^)    perpen- 
dicular to  line    (1),    and 
(9i?2(  =^2)  perpendicular  to  line  (2) .     Then  d  =  QP^  =p^  —py 
The  equation  of  a  line  parallel  to  line  (1)  is  of  the  form 
3a:  +  'iz/-f^=0;  this  will  represent  line  (2)  itself  if  k  be 
so  determined  that  the  line   shall  j)ass  through  the   point 

Pj=(2,  I),  i.e,,ii  3-2h-4.|  +  A:  =  0,  i.e.,  if  ^  =  -  12. 

The  equation  of  line  (2)  is  then 

3a;  +  4z/-12  =  0 

Therefore  [by  Art.  58,  (3)  or  (4)] 

12  12         ,  7  7 

— .    and   r>.  = 


(2) 


P2  = 


4-V4^  +  3-^ 


12 

=  =  -,   and;>i  = 


+  V42  +  32     5' 


hence  the  required  distance  is  t?=  QP^  = 


12-7 


=  1. 


Similarly,  in  general,  to  find  the  distance  of  any  given 
point  P^  =  (a;^,  ^/j)  from  any  given  line 


Ax-hBi/-\-C=0 


(1) 


let  line  (1)  be  the  locus  of  equation  (1)  and  let  P^  be  the 
given  point.  The  equation  of  a  line  parallel  to  (1)  is  of 
the   form   Ax  ■{-  Bi/ -\- K=0 ;   this  will   be   the   line   (2)   if 


64.]  THE  STRAIGHT  LINE  107 

Ax^  +  %i  +  K=  0,  i.e.,  if  K=  -  (Ax^  +  Bi/^).     The  equa- 
tion of  line  (2)  is  then 

Ax-\-Bi/-(Ax^i-Bi/^)=0.      ...      (2) 
Therefore     /?„  =  — ^i  "^     '^^,    »  =  —  ~ 

wherein  the  sign  of  the  radical  is  to  be  chosen  in  accord 

with  Art.  58  (3); 

hence  ^^Ax,+Bp,^C^       ^       ^       ^       .^g-. 

If  the  equation  of  the  given  line  is  so  written  that  its 
second  member  is  zero,  this  formula  may  be  translated  into 
words  thus  :  To  get  the  distance  of  a  given  point  from  a  given 
line,  write  the  first  member  of  the  equation  alone,  substitute 
for  the  variables  therein  the  coordinates  of  the  given  point, 
and  divide  the  result  by  the  square  root  of  the  sum  of  the 
squares  of  the  coefficients  of  x  and  y  in  the  equation,  —  the 
sign  of  this  square  root  being  chosen  opposite  to  that  of 
the  number  represented  by  C. 

If,  in  formula  [16],  d  is  positive,  then  p^> Pi,  and  P^ 
and  the  origin  are  on  opposite  sides  of  the  given  line  ;  if 
d  is  negative,  P2<Pv  ^^^  -^i  ^^^  ^^®  origin  are  on  the 
sa77ie  side  of  the  given  line. 

EXERCISES 

1.   Find  the  distance  of  the  point  (2,  -7)  from  the  hne  3a:  — 62/4-1  =  0. 
By  formula  [16],   d  =  ^•^-^C-7)+l  _  _    49  . 


-V32  +  62  3^5 

This  result,  besides  giving  the  numerical  value  of  the  distance,  shows 
also  that  the  point  (2,  -7)  and  the  origin  are  on  the  same  side  of  the 
line  Qx  —  6y  -}-  1  =0. 

2.  Find  the  distance  of  the  point  (4,  5)  from  the  line  iy  +  ox  =  20. 

3.  Find  the  distance  of  the  point  (2,  7)  from  the  line  3  ?/  -  2x  =  17. 


108  ANALYTIC  GEOMETRY  [Ch.  V. 

4.  Find  the  distance  of  the  point  (a,  b)  from  the  line  -  +  y  =  1. 

5.  Find  the  distance  of  the  intersection  of  the  two  lines,  y+4.  =  3x 
and  6x=  7/  —  2,  from  the  line  2  ?/  —  7  =  9.  On  which  side  of  the  latter 
line  is  the  point  ? 

6.  Find   the   distance   of    the   point    of    intersection    of   the   lines 

1  ?/  —  5 

2  a;  — 5y  =  ll    and    4  a;  =  3  w  +  15   from   the   line-a;  +  ^ — ^=6.      On 

2  4 

which  side  of  the  latter  line  is  the  point?    Plot  the  figure. 

7.  How  far  is  the  point  ("6,  -1)  from  3y  =  7x  +  S'^    On  which  side? 

8.  By  the  method  of  Art.  64,  find  the  distance  of  the  origin  from 
the  line  5x  —  2y  =  7  ;  also  from  the  line  Ax  +  By  +  C  =  0.  Check  the 
results  by  Art.  58  (3). 

9.  Find  the  distance  of  the  point  ("4,  -5)  from  the  line  joining  the 
two  points  (3,  -1)  and  (~4,  2).     On  which  side  is  it? 

10.  Find  the  distance  of  the  point  (xj,  y^)  from  the  line  y  =  mx  +  h. 

11.  Find  the  altitudes  of  the  triangle  formed  by  the  lines  whose  equa- 
tions are  x  +  ?/  +  1  =  0,  3  a;  +  5  ?/  +  11  =  0,  and  a;  +  2?/  +  4=:0.  Check 
the  result  by  finding  the  area  of  the  triangle  in  two  ways. 

12.  Show  analytically  that  the  locus  of  a  point  which  moves  so  that 
the  sum  of  its  distances  from  two  given  straight  lines  is  constant  is  itself 
a  straight  line. 

13.  Express  by  an  equation  that  the  point  P^  =  (a;^  y^)  is  equally 
distant  from  the  two  lines  2x  —  y=  11  and  4  a;  =  3?/ +  5.  (Give  two 
answers.)  Should  P^  move  in  such  a  way  as  to  be  always  equidistant 
from  these  two  lines,  what  would  be  the  equation  of  its  locus  ? 

14.  Find,  by  the  method  of  exercise  13,  the  equations  of  the  bisectors 
of  the  angle  formed  by  the  lines  3  a:  +  4  ?/  =  12  and  4  a;  +  3  ?/  =  24. 

65.     Bisectors  of  the  angles  between  two  given  lines.     The 

bisector  of  an  angle  is  the  locus  of  a  point  which  moves 

so  that  it  is  always  equally  distant  (numerically)  from  the 

sides  of  the  angle.     From  this  property  its  equation  may 

easily  be  found. 

E.g.^  find  the  equations  of   the  bisectors  of  the  angles 

between  the  lines 

3a;  +  4i/-l  =  (>,      ...  (1) 

and  12a;-53/  +  6  =  0.      .     .     .  (2) 


64-65.] 


THE  STRAIGHT  LINE 


109 


Let  Pj  =  (a;j,  ?/j)  be  any  point 
on  the  bisector  (3). 

Then  Q^P^  =  -R^P^  [since  0 
and  P^  are  on  opposite  sides  of 
line  (1)  and  On  the  same  side  of 
(2)  ;  or  vice  versa]. 

But  np       3»,+4.y^-l 

+  V32  +  42 

=  ^^^  +  \y,-\  (Art.  64), 


Y 

/ 

\- 

.R, 

r 

"--^.^       (3) 

A 

\ 

X 

/ 

\ 

0  ^ 

X 

m 

^ 

Vi) 

\a) 

Fig,  50. 


and 


jip  ^  12  rr^  -  5  ,y,  +  6  ^  12  rr,  -  5  ,yT  +  6  . 

^     ^  _Vl22-i-.^2  -13 


Hence 


(5) 
(6) 


VI22  +  52 

.    3a;^  +  4yT-l_12:ri-5,y, +6. 
5  "  13 

t.e.,  21 0^1  -  77  ?/i  +  43  =  0.  .     .     , 
21 2; -77?/ +  43  =  0      .     .     . 

is  the  equation  of  the  bisector  (3),  for  equation  (5)  asserts 
that  if  (a^j,  1/1)  be  the  coordinates  of  any  point  on  this  bisec- 
tor they  satisfy  equation  (6). 

Similarly,  let  P^  =  (^,  k)  be  any  point  on  line  (4),  the 
other  bisector,  then  $2^2  —  -^2^2  [since  0  and  P^  are  on 
opposite  sides  of  the  lines  (1)  and  (2),  or  else  both  on  the 
same  side  of  each  of  these  lines] ; 

Sh-h4:k-l  12h-bk-h6 

5  ~  13  ' 

99A  +  27Aj  +  17  =  0.  .     .     .  (7) 

99a: +  27y +  17  =  0   ...  (8) 

is  the  equation  of  the  bisector  (4),  for  the  same  reason  as 
given  above. 


I.e., 
Hence 


110  ANALYTIC  GEOMETBY  [Ch.  V. 

Geometrically  it  is  well  known  that  two  such  bisectors, 
(3)  and  (4),  are  perj^endicular  to  each  other  :  their  equa- 
tions, also  prove  that  fact. 

The  equations  of  the  bisectors  of  the  angles  between  any 
two  lines,  as  A^x  +  B-^y  +  (7^  =  0  and  A^^x  +  B^y  +  C\  =  0, 
are  found  in  precisely  the  same  way  as  that  employed  in  the 
numerical  example  just  considered. 

EXERCISES 

1.  Find  the  equations  of  the  bisectors  of  the  angles  between  the  two 
lines  X  —  y  +  Q  =  0  and  — ^ —  =  oy  —7. 

2.  Show  that  the  line  llx-^^y  +  l  =  0  bisects  one  of  the  angles 
between  the  two  lines  12  x  -  5 y  +  7  =  0,  and  3  x  -^  4: y  -  2  =  0.  Which 
angle  is  it  ?     Find  the  equation  of  the  bisector  of  the  other  angle. 

3.  Show  analytically  that  the  bisectors  of  the  interior  angles  of  the 
triangle  whose  vertices  are  the  points  (1,  2),  (5,  3),  and  (4,  7)  meet  in  a 
common  point. 

4.  Show  analytically,  for  the  triangle  of  Ex.  3,  that  the  bisectors  of 
one  interior  and  the  two  opposite  exterior  angles  meet  in  a  common 
point. 

5.  Find  the  angle  from  the  line  3a:4-2/  +  12=:0to  the  line  ax  +  by 
+  1  =  0,  and  also  the  angle  from  the  line  ax  -\-  by  +  1  =  0  to  the  line 
X  +  2y  -1  =0. 

By  imposing  upon  a  and  b  the  two  conditions :  (1)  that  the  angles 
just  found  are  equal,  and  (2)  that  the  line  ax  -\-  by  -}-  1  =  0  passes  through 
the  intersection  of  the  other  two  lines,  determine  a  and  b  so  that  this  line 
shall  be  a  bisector  of  one  of  the  angles  made  by  the  other  two  given 
lines. 

66.   The  equation  of  two  lines.     By  the  reasoning  given  in 

Art.  40,  it  is  shown  that  if  two  straight  lines  are  represented 

by  the  equations 

.     A^x-hB^y^C^  =  0        ,       .       .       (1) 

and  A^x  +  B^y  +  (72  =  0,       .       .       .       (2) 

then  both  these  lines  are  represented  by  the  equation 

iA^x  +  B^y  +  C{)(iA^x  +  B^  +  C^)  =  0;    .    .    .  (3) 


65-67.]  THE  8TBA1GHT  LINE  111 

^.e.,  two  straight  lines  are  here  represented  by  an  equation 
of  the  second  degree. 

Conversely,  if  an  equation  of  the  second  degree,  whose 
second  member  is  zero,  can  have  its  first  member  separated 
into  two  first  degree  factors,  with  real  coefficients,  as  in 
equation  (3),  then  its  locus  consists  of  two  straight  lines. 

Thus  the  equation 

may  be  written  in  the  form 

(2a;-3^  +  7)(a;+2/-f-l)  =  0, 

which  shows  that  it  is  satisfied  when  2a;  —  3?/+7  =  0,  and 
also  when  x  -\-  y  -\-  1  =  0.  Its  locus  is  therefore  composed 
of  the  two  lines  whose  equations  are  : 

2a;-3y  +  7  =  0,  and  x  +  y  -[-1  =  0. 

67.  Condition  that  the  general  quadratic  expression  may  be 
factored.  The  most  general  equation  of  the  second  degree 
between  two  variables  may  be  written  in  the  form 

Ax^+2Exy^-By'^+'iax-^2Fy +0=0.  ,     .     .     (1) 

It  is  required  to  find  the  relation  that  must  exist  among  the 
coefficients  of  this  equation  in  order  that  its  first  member 
may  be  separated  into  two  rational  factors,  each  of  the  first 
degree,  i.e.,  it  is  required  to  find  the  condition  that  the  equa- 
tion may  be  written  thus  : 

(^ct^x  -f  h^y  +  c^{a^x  +  h^y  -{-c^^=0.    .     .     .      (2) 

Evidently  if  equation  (1)  can  be  written  in  the  form  of 
equation  (2),  then  the  values  of  x  obtained  from  equation 
(1)  are  rational,  and  are  either 

^^-^^-hy  or  a;  =  - ^2 - hy, 

H  «2 


112  ANALYTIC  GEOMETRY  [Ch.  V. 

Solving  equation  (1)  for  x  in  terms  of  ?/,  by  completing 
the  square  of  the  2;-terms,  it  becomes 

.  ^v  +  2  ^(%  +  a)x + {iTy  +  ay 

=  -  ABf-2AFi/  -AQ  +  {Hy  +  ay, 
i.e.,  Ax-\-Hy+  a 

=  V(^2  _  AB)y'^  -  2  {Ha  -  AF)y  +  a^-AO, 
and  finally, 


H     a  ,  1 


x=---y--±--VCE'^-AB)y''-2{Ra-AF}y+a^-A0. 

A  Jx      JL 

But  since  x  is,  by  hypothesis,  expressible  rationally  in 
terms  of  y,  therefore  the  expression  under  the  radical  sign 
is  a  perfect  square,  and  therefore 

CHa  -  AFy-^H'^-  AB)(a^  -  ao)=  o, 

i.e.,  ABC  +  2  FGH-  AF^  -  BG^  -  CH^  =  0.     .      .      [17] 

If  this  condition  among  the  coefficients  is  fulfilled,  then 
equation  (1)  has  for  its  locus  two  straight  lines. 

The  expression  AB0+2FaR -  AF^  -  Ba'^  -  CR^  is 
called  the  discriminant  of  the  quadratic,  and  is  usually 
represented  by  the  symbol  A. 

Note.  The  analytic  work  just  given  fails  if  ^  =  0.  In  that  case 
equation  (1)  may  be  solved  for  y  instead  of  solving  it  for  x,  and  the  same 
condition,  viz.  A  =  0,  results.  If,  however,  both  A  and  B  are  zero,  then  the 
above  method  fails  altogether.     In   that  case  equation  (1)  reduces  to 

2Hxy  +  2Gx  +  2Fy+C  =  0 (3) 

If  the  first  member  of  equation  (3)  can  be  factored,  then  evidently  the 
equation  must  take  the  form 

{ax  +  b)(cy  +  d)  =  Q     .......     (4) 

which  shows  that  equation  (3)  is  satisfied  for  all  values  of  y  provided 

X  =  — ,  a  constant.    Let be  represented  by  k,  then  equation  (4) 

a  a 

becomes  2  Hky  +  2  Gk +  2  Fy  +  C  =  0, 

i.e.,  2iHk  +  F)y  +  2Gk+C  =  0, 


67.]  THE  STRAIGHT  LINE  113 

and  is  satisfied  for  all  values  of  y ; 

Hk-^  F=0,    and    2Gk-{-C  =  0; 
hence,  eliminating  k,  2  FG  —  CH  =  0. 

But  this  is  the  expression  to  which  A  reduces  when  A  =  B  =  0  and 
H  =^0]  hence,  in  all  cases,  A  =  0  is  the  necessary  condition  that  the 
above  quadratic  may  be  factored. 

That  A  =  0  is  also  the  sufficient  condition  is  readily  seen  by  retracing 
the  steps  from  equation  [17]  when  at  least  one  of  the  coefficients  .4,  B 

differs  from  zero.     But  it  is  also  sufficient  when  ^4  =  B  =  0]  for,  in  that 

F   G       C 
case,  A  =  0  becomes  2  FG  —  CH  =  0,  which  may  be  written  — .  —  =  — — . 
X  H  H     2  H 

Under  the  same  circumstances  equation  (1)  becomes  equation  (3),  which 
may  be  written 

xy+^xi-^y+-^  =  0 (4) 

-^      H        H^      2H  ^  ^ 

F   G  G 

Substituting for  in  equation  (4),  it  becomes 

H  H         2  H 

3:y  +  ^x  +  ^V  +  ^-^  =  0 (5) 

^      H        H^      H  H  ^  ^ 

(.  +  |)(.  +  |)=o, 

which  establishes  the  sufficiency  of  the  condition  for  this  case  also. 

To  illustrate   the   use  of   equation    [17]*   examine  the  equation   of 

Art.  66 : 

2x^  -  xy  -oy^  +  Qx  +  ^y  +  1  =0. 


*  As  an  illustration  of  another  practical  method  of  factoring  a  quadratic 

expression,  lohen  factoring  is  possible-,  i.e.,  if  equation  [17]  holds,  find  the 

factors  of 

2x^  -Ixy  -15y'^  +  7x-l'ly  -4t. 

Factor  the  terms  free  from  y, 

2a;2  +  7x-4=(2a;-l)(a;  +  4); 

factor  the  terms  free  from  x, 

-  152/2  -  17  y  -  4=  (3  ?/  -  1)(_  5y  +  4); 

combine  the  factors  containing  the  same  constant  term, 

(2x  +  Sy-l),    (x-6y+4:); 

these  will  be  the  factors  of  the  given  quadratic  expression. 

TAN.  AN.  GEOM.  —  8 


114  ANALYTIC  GEOMETRY  [Ch.  V. 

Here      ^  =  2,  5  =  -  3,  C  =  7,  i^  =  -  |,    G=|,andF  =  2; 

hence  a  =  -  42  -  9  -  8 +^  -  7  =  0; 

4       4 

therefore  the  first  member  can  be  factored. 

The  factors  may  be  found  as  follows :  transposing,  dividing  by  2,  and 

completing  the  square  of  the  a;-terms,  the  equation  may  be  written  in 

the  form         x^  +  ^^  +  (^)'  "^ll^^'  -  2^/  +  1); 

therefore  the  given  equation,  divided  by  2,  may  be  written  in  the  form, 

i.e.,  (a;  +  3/  +  1)  (a;  -  I  ?/  +  0  =  0  ; 

hence  the  locus  of  the  original  equation  consists  of  the  straight  lines 

a:  +  ?/  +  1  =  0  and  2a;-33/  +  7  =  0, 
which  agrees  with  the  result  of  Art.  66. 

EXERCISES 

Prove  that  the  following  equations  represent  pairs  of  straight  lines  ; 
find  in  each  case  the  equations  of  the  two  lines,  the  coordinates  of  their 
point  of  intersection,  and  the  angle  between  them. 

1.  6  y2  _  x^  -  a:2  -f  30  ?/  +  36  =  0. 

2.  a:2-2a:?/-3  3/2  +  2a;-23/  +  l=0. 

3.  x^  —  2  x?/  sec  a  +  2/2  =  0. 

4.  a;2  +  6  a;?/  +  9  ?/2  +  4  a:  +  12  ?/  -  5  =  0. 

5.  For  what  value  of  k  will  the  equation 

a:2  _  3  ^^  ^  2/2  +  10  a:  -  10  2/  +  Z:  =  0 
represent  two  straight  lines  ? 

Suggestion  :  Place  the  discriminant  (A)  equal  to  zero,  and  thus  find 
A;  =  20. 

Find  the  values  of  k  for  which  the  following  equations  represent  paii-s 
of  straight  lines.  Find  also  the  equation  of  each  line,  the  point  of  inter- 
section of  each  pair  of  lines,  and  the  angle  between  them. 

6.  6  a;2  +  2  kxy  +  12  ?/2  +  22  a:  +  31  ?/  +  20  =  0. 

7.  12  a;2  +  36  a;?/  +  A;2/2  +  6  a;  +  6  ^  +  3  =  0. 


67-68.] 


THE  STRAIGHT  LINE 


115 


8.  4iX^-12xy  ^Qy"^  -  kx  -{-6y  +1  =  0. 

9.  The  equations  of  the  opposite  sides  of  a  parallelogram  are 

a:2  _  7  a;  +  6  =  0  and  1/2  -  14  2/  +  40  =  0. 
Find  the  equations  of  the  diagonals. 

10.  Find  the  conditions  that  the  straight  lines  represented  by  the  equa- 
tion Ax"^  +  2  Bxy  +  Cy^  =  0  may  be  real ;  imaginary;  coincident ;  perpen- 
dicular to  each  other. 

11.  Show  that  6  x2  +  5x2/  -  6  2/2  =  0  is  the  equation  of  the  bisectors  of 
the  angles  made  by  the  lines  2  x^  +  12  xy  +  7  y"^  =  0.  Does  the  first  set 
of  lines  fulfil  the  test  of  exercise  10  for  perpendicularity? 

68.  Equations  of  straight  lines :  coordinate  axes  oblique. 
Since  in  the  derivation  of  equations  [9]  and  [10]  (Arts.  51 
and  52)  only  properties  of  similar  triangles  were  employed, 
therefore  these  two  equations  are  true  whether  the  coordi- 
nate axes  are  rectangular  or  oblique. 

The  other  three  standard  forms  however,  viz.  ?/  =  mx  +  5, 
y  —  y^  =  m(x  —x{)^  and  x  cos  a  +?/  sin  a=p^  the  derivation  of 
which  depends  upon  right  triangles,  are  no  longer  true  if 
the  axes  are  inclined  to  each  other  at  an  angle  «  ^  ^-    Equa- 

tions  which  correspond  to  these,  but  which  are  referred  to 
oblique  axes,  will  now  be  derived. 

(1)  Equation  of  straight  line  through  a  given  point  and  in 
a  given  direction.  Let  LL^  be  the  straight  line  through  the 
fixed  point  P^=(x^^  y^  and 
making  an  angle  6  with  the 
ic-axis,  let  P  =  (a;,  y^  be  any 
other  point  on  XZ/j,  and  let 
CO  be  the  angle  between  the 
axes. 

Draw  P^R  parallel  to  the 
a;-axis,  also  draw  the  ordinates  M^P^  and  MP.     Then 

6=ZXAL=:ZRP^L    and    ZP^PR=(o-6. 


116  ANALYTIC  GEOMETRY  [Ch.  V. 

Hence         -— -^  =  -r— 7 -xr-        [law  of  smesj 

Pji^      sin(ft)-^) 

Substituting  in  this  equation  the  coordinates  of  P^  and  P, 

it  becomes 

y  -  yi^       sin  ^ 
0^  —  a?!      sin  (w  —  ^)' 

sin  Or  ^  n  Qi 

which  is  the  required  equation. 

When  o)  =  -  this  equation  reduces  to  equation  [11],  ^.e., 
to  y  —  yi=  'rn  {x  —  a^i),  where  m  =  tan  6  ;  but  it  must  be 
observed  that  if  «  =?^^,  then  the  coefficient  of  x  in  equation 

[18]  does  not  represent  the  slope  of  the  line.     If,  however, 
the  slope  of  the  line   [18],  i.e.^  the  tan  6  for  this  line,  is 

desired,  it  is  easily  found  thus  :  let    =  ^,   from 

^  sin  (a)-6>) 

which  is  obtained  tan  6  = . 

1  -\-  k  cos  (o 

If,  in  the  derivation  of  equation  [18],  the  given  point  is 

that  in  which  the  line  LL-^  meets  the  2/-axis,  ^.e.,  if  Pi  =(0,  i), 

then  equation  [18]  reduces  to 

y=  .   7^  ..^  +  h,     ■      •      •     [19] 

sm  (o)  —  6) 

which  corresponds  to  equation  [12],  but  the  coefficient  of  x 
is  not  the  slope  of  the  line. 

(2)  Equation  of  a  straight  line  in  terms  of  the  perpendic- 
ular upon  it  from  the  origin,  and  the  angles  which  this  perpe7i- 
dicular  inakes  with  the  axes. 


68.]  THE   STRAIGHT  LINE  117 

Let  LL^  be  the  straight  line  whose  equation  is  sought, 
let  the  perpendicular  from  the 
origin  upon  it  (01^=  p)  make 
the  angles  a  and  ^  respectively 
with  the  axes,*  and  let  P= 
(a:, «/)  be  any  point  on  LLi. 

Draw  the  ordinate  MP  ;  then,       /  fig.52. 

by  Art.  17, 

OM  cos  a  +  MP  cos  /3  =  ON, 

i.e.^  X  cos  a  -\-  y  cos  ^  =  p^       .      .       .       [20] 

which  is  the  required  equation. 

If  ft)  is  the  angle  between  the  axes,  then  /3  =  &>  —  a,  and 
equation   [20]  may  be  written  x  cos  a  ■\-  y  cos  (o)  —  a)  =Jt?. 

If  ft)  =  -  ,  then  this  equation  reduces  to  x  cos  a  -Vy  sin  a-=  p, 

Li 

which  agrees  with  equation  [13]. 

EXERCISES 

1.  The  axes  being  inclined  at  the  angle  60°,  find  the  inclination  of 

the  line  ?/  =  2  a:  +  5  to  the  a:-axis. 

2.  The  axes  being  inclined  at  the  angle  -,  find  the  angles  at  which 
the  lines  32/  +  7a:-l=0  and  a:  +  ?/  +  2  =  0  cross  the  a:-axis. 

3.  Find  the  angle  between  the  lines  in  exercise  2. 

4.  The  center  of  an  equilateral  triangle  of  side  6  is  joined  by  straight 
lines  to  the  vertices.  If  two  of  these  lines  are  taken  as  coordinate  axes, 
find  the  coordinates  of  the  vertices,  and  the  equations  of  the  sides. 

5.  Prove  that  for  every  value  of  w,  the  lines  a;  +  2/  =  c  and  x  —  y  =  d 
are  perpendicular  to  each  other. 


*  The  angles  a  and  j3  are  the  direction  angles  of  the  line  ON,  and  their 
cosines  are  the  direction  cosines  of  that  line. 


118 


ANALYTIC  GEOMETRY 


[Ch.  V. 


69.   Equations  of  straight  lines:  polar  coordinates. 

(1)    Line  through  two  given  points.     Let  OR  be  the  initial 

line,  0  the  pole,  Pi 
=  (pi,  ^i)^  and  P2  = 
(/02,  62)-,  the  two  given 
points,  and  let  P  = 
(/3,  6)  be  any  other 
point  on  the  line 
through  Pi  and  Pg. 


Fig. 53. 


Then  (if  A  stands  for  '  area  of  triangle ') 

A  OPiP,  =  A  OPiP  +  A  OPP2. 
i.e.,     I  piP2sin(^2-^i)  =  J/o/?isin(<^-<^i)+  I P2P  sin  (^2-^)^ 
hence  ppi  sin  (^—  ^1)  4-/3ip2  sin  (^1  —  ^2) 

+  /92psin(i92-6>)  =  0.*.     .     .     [21] 
This  equation  may  also  be  written  in  the  form 

sin  {Oi  -  62)  I  sin  (jO^  -  6)  ^  sin  {0  -  Oi)  ^^  ,, 
P  Pi  P2 

(2)  Equation  of  the  line  in  terms  of  the  perpendicular  upon 
it  from  the  jjole^  and  the  angle  which  this  perpendicular  makes 
with  the  initial  line.  Let  OR  be  the  initial  line,  0  the  pole, 
and  LK  the  line  whose  equation  is 
sought.  Also,  let  N=  (p,  a)  be  the 
foot  of  the  perpendicular  from  0 
upon  LK.,  and  let  P=(p.,  6)  be  any 
other  point  on  LK.     Draw  ON  and 

OP  ;  then 

ON 


^^=  cos  NOP, 

i.e.,  p  cos  (^  —  a)  =  p, 

which  is  the  required  equation. 


[22] 


*  Observe  the  symmetry  here  ;  cf .  foot-note,  Art.  29. 


69.]  THE  STRAIGHT  LINE  119 

EXERCISES 

1.  Construct  the  lines  : 

(a)     p  cos  ((9- 30°)  =  10;  (c)    pcos^^-|^=9; 

(&)     p  sin  0  =  2',  (d)     p  cos  {6  -  tt)  =  Q. 

2.  Find  the  polar  equations  of  straight  lines  at  a  distance  3  from  the 
pole,  and:  (1)  parallel  to  the  initial  line;  (2)  perpendicular  to  the  initial 
line. 

3.  A  straight  line  passes  through  the  points  (5,  -45°)  and  (2,  90°) ; 
find  its  polar  equation. 

4.  Find  the  polar  equation  of  a  line  passing  through  a  given  point 
(/Op  Oi)  and  cutting  the  initial  line  at  a  given  angle  <f>  =  t3in~'^k. 

5.  Find  the  polar  coordinates  of  the  point-  of  intersection  of  the  lines 

p  cos  ( ^  —  ^  J  =  2  a,  p  cos  (  ^  -  ^  J  =  a. 

EXAMPLES  ON  CHAPTER  V 

1.  The  points  ("1,  2)  and  (3,  -2)  are  the  extremities  of  the  base  of 
an  equilateral  triangle.  Find  the  equations  of  the  sides,  and  the  coordi- 
nates of  the  third  vertex.     Two  solutions. 

2.  Three  of  the  vertices  of  a  parallelogram  are  at  the  points  (1,  1), 
(3,  4),  and  (5,  -2).  Find  the  fourth  vertex.  (Three  solutions.)  Find 
also  the  area  of  the  parallelogram. 

3.  Find  the  equations  of  the  two  lines  drawn  through  the  point  (0,  3), 
such  that  the  perpendiculars  let  fall  from  the  point  (6,  6)  upon  them  are 
each  of  length  3. 

4.  Perpendiculars  are  let  fall  from  the  point  (5,  0)  upon  the  sides  of 
the  triangle  whose  vertices  are  at  the  points  (4,  3),  (-4,  3),  and  (0,  ~5). 
Show  that  the  feet  of  these  three  perpendiculars  lie  on  a  straight  line. 

Find  the  equation  of  the  straight  line 

5.  through  the  origin  and  the  point  of  intersection  of  the  lines 
X  —  y  =  4:  and  7  x  -{-  y  -]-  20  =  0.  Prove  that  it  is  a  bisector  of  the  angle 
formed  by  the  two  given  lines. 

6.  through  the  intersection  of  the  lines  3x  —  4?/  +  l=0  and 
5x  +  y  =  1,  and  cutting  off  equal  intercepts  from  the  axes. 

7.  through  the  point  (1,  2),  and  intersecting  the  line  a;  +  y  =  4  at  a 
distance  |  V6  from  this  point. 


120  ANALYTIC  GEOMETRY  [Ch.  V. 

8.  Find  the  equation  of  a  straight  line  through  the  point  (4,  5)  and 
making  equal  angles  with  the  lines  3 x  =  4  ?/  +  7  and  by  =  12x  -\-  Q. 

9.  Prove   analytically  that  the  diagonals  of  a  square  are  of  equal 
length,  bisect  each  other,  and  are  at  right  angles. 

10.  Prove  analytically  that  the  line  joining  the  middle  points  of  two 
sides  of  a  triangle  is  parallel  to  the  third  side  and  equal  to  half  its  length. 

11.  Find  the  locus  of  the  vertex  of  a  triangle  whose  base  is  2  a  and 
the  difference  of  the  squares  of  whose  sides  is  4  c^.     Trace  the  locus. 

12.  Find  the  equations  of  the  lines  from  the  vertex  (4,  3)  of  the  tri- 
angle of  Ex.  4,  trisecting  the  opposite  side.  What  are  the  ratios  of  the 
areas  of  the  resulting  triangles  ? 

13.  A  point  moves  so  that  the  sum  of  its  distances  from  the  lines 
^-3x4-11  =  0  and  7  x  -2y  +  I  =  0  is  6.  Find  the  equation  of  its 
locus.     Draw  the  figure. 

14.  Find  the  equation  of  the  path  of  the  moving  point  of  Ex.  13,  if 
the  distances  from  the  fixed  lines  are  in  the  ratio  3 : 4. 

15.  Solve  examples  13  and  14,  taking  the  given  lines  as  axes. 

16.  The  point  (2,  9)  is  the  vertex  of  an  isosceles  right  triangle  whose 
hypotenuse  is  the  line  dx  —  7 y  =  2.  Find  the  other  vertices  of  the 
triangle. 

17.  The  axes  of  coordinates  being  inclined  at  the  angle  60°,  find  the 
equation  of   a  line  parallel   to  the  line  x  -^  y  =  3  a,  and   at  a  distance 

aV3  ^ 
^2     from  it. 

18.  Find  the  point  of  intersection  of  the  lines 

P  = 


-P :-    and    pcos(^--)  =  a. 


cos    a  — 


For  what  value  of  &,  in  each  line,  is  p  =  oo  ?    At  what  angles  do  these  lines 
cut  their  polar  axes?    Find  the  angle  between  the  lines.     Plot  these  lines. 

19.   Find  the  equation  of  a  straight  line  through  the  intersection  of 
y  —  7x  —  4:  and  2x  -\-  y  =  5,  and  forming  with  the  x-axis  the  angle  -• 


20.   Find  the  equation  of  the  locus  of  a  point  which  moves  so  as  to  be 
always  equidistant  from  the  points  (2,  1)  and  (~3,  ~2). 


69.]  THE  STRAIGHT  LINE  121 

21.  Find  the  equation  of  the  locus  of  a  point  which  moves  so  as  to  be 
always  equidistant  from  the  points  (0,  0)  and  (3,  2).  Show  that  the 
points  (0,  0),  (3,  2),  and  (1,  -1)  are  the  vertices  of  an  isosceles  triangle. 

22.  Find  the  center  and  radius  of  the  circle  circumscribed  about  the 
triangle  whose  vertices  are  the  points  (2,  1),  (3,  -2),  (-4,  -1). 

23.  Find  analytically  the  equation  of  the  locus  of  the  vertex  of  a 
triangle  having  its  base  and  area  constant. 

24.  Prove  analytically  that  the  locus  of  a  point  equidistant  from  two 
given  points  (x^,  y^  and  {x^,  2/2)  is  the  perpendicular  bisector  of  the  line 
joining  the  given  points. 

25.  The  base  of  a  triangle  is  of  length  5,  and  is  given  in  position ; 
the  difference  of  the  squares  of  the  other  two  sides  is  7 ;  find  the  equa- 
tion of  the  locus  of  its  vertex. 

26.  AVhat  lines  are  represented  by  the  equations : 

(a)    x'^y  =  xy^ ;       (^)    14  x^  —  5 a:?/  —  y^  =  0  ;      (y)    xy  =  0? 

27.  What  must  be  the  value  of  c  in  order  that  the  lines  3x-{-y  —  2  =  0, 
2a:— !/  —  3  =  0,  and  5x  +  2y  +  c  =  0  shall  pass  through  a  common  point  ? 

28.  By  finding  the  area  of  the  triangle  formed  by  the  three  points 
(3  a,  0),  (0,  3  6)  and  (a,  2  b),  prove  that  these  three  points  are  in  a  straight 
line.  Prove  this  also  by  showing  that  the  third  point  is  on  the  line  join- 
ing the  other  two. 

29.  Find,  by  the  method  of  Art.  39,  the  point  of  intersection  of  the 
two  lines  2  x  —  Sy  -{-7  =  0  and  4  a:  =  6  ?/  +  2  ;  and  interpret  the  result 
by  means  of  Arts.  41  and  60. 

30.  Prove  by  Art.  10  (cf.  also  Arts.  41  and  60),  that  the  equations  of 
two  parallel  lines  differ  only  in  the  constant  term. 

31.  Find  the  equations  of  two  lines  each  drawn  through  the  point 
(4,  3),  and  forming  with  the  axes  a  triangle  whose  area  is  8. 

32.  Find  the  equation  of  a  line  through  the  point  (2,  -5),  such  that 
the  portion  between  the  axes  is  divided  by  the  given  point  in  the  ratio 
7:5. 

33.  Find  the  equation  of  the  perpendicular  erected  at  the  middle 
point  of  the  line  joining  (5,  2)  to  the  intersection  of  the  two  lines 

x  +  2y  =  ll     and     9x-2y=59. 


122  ANALYTIC  GEOMETRY  [Ch.  V.  69. 

34.  A  point  moves  so  that  the  square  of  its  distance  from  the  origin 
equals  twice  the  square  of  its  distg^nce  from  the  x-axis ;  find  the  equation 
of  its  locus. 

35.  Given  the  four  lines 

x-2y  +  2  =  0,  X  +  2ij  -2  =  0,  dx-y  -S  =  0  and  a;  +  ?/+6  =  0; 
these  lines  intersect  each  other  in  six  points ;   find  the  equations  of  the 
three  new  lines  (diagonals),  each  of  which  is  determined  by  a  pair  of  the 
above  six  points  of  intersection. 

36.  Find  the  points  of  intersection  of  the  loci : 

(a)     p  cos  ( ^  -  ^  J  =  a  and  p  cosf  ^  -  ^  j  =  a ; 

(13)    pcos(0-f\  =  ^  Siud  p  =  asme. 

If  two  sides  of  a  triangle  are  taken  as  axes,  the  vertices  are  (0,  0), 
(Xj,  0),  (0,  2/2)-     Prove  analytically  that: 

37.  the  medians  of  a  triangle  meet  in  a  point ; 

38.  the  perpendicular  from  each  vertex  to  the  opposite  sides  meet 
in  a  point; 

39.  the  line  joining  the  middle  points  of  two  sides  of  a  triangle  is 
parallel  to  the  third  side. 

40.  Show  that  the  equation  56  x^  -  441  xy  -  oQy^  -79  x  -  47  ?/  +  9  =  0 
represents  the  bisectors  of  the  angles  between  the  straight  lines  repre- 
sented by  15x2-  16a:?/ -482/2- 2a;  +  16?/-  1=0. 

41.  Two  lines  are  represented  by  the  equation 

.4x2  +  2  Hxy  +  %2  =  0. 

Find  the  angle  between  them. 


CHAPTER   VI 


TRANSFORMATION  OF  COORDINATES 


70.   That  the  coordiuates  of  a  point  which  remains  fixed 
in  a  plane  are  changed  by  changing  the  axes  to  which  this 
fixed  point  is  referred,   is  an  immediate 
consequence   of  the  definition  of  coordi- 
nates. 

It  is  also  evident  that  the  different 
kinds  of  coordinates  of  any  given  point 
(Cartesian  and  polar,  for  example)  are 
connected  by  definite  relations  if  the  ele- 
ments of  reference  (the  axes)  are  related  in  position.  E.g.i, 
the  point  §,  when  referred  to  the  polar  axis  OX  and  the  pole 
0,  has  the  coordinates  (5,  30°),  but  when  it  is  referred  to 

the  rectangular  axes  OX  and 
0!F  the  coordinates  of  this  same 
point  are  (f  V3,  |)  ;  and  gen- 
erally, if  (/?,  ^)  be  the  co- 
ordinates of  a  point  when 
referred  to  OX  and  0,  then 
(/o  cos  ^,  p  sin  0')  are  its  coordi- 
nates when  it  is  referred  to  the 
rectangular  axes  OX  and  OY. 

Again :   while  a  curve  remains  fixed  in  a  plane,  its  equa- 
tion may  often  be  greatly  simplified  by  a  judicious  change  of 

123 


r 


.0' 


X 


Fig.  56 


124 


ANALYTIC  GEOMETRY 


[Ch.  VI. 


the  axes  to  which  it  is  referred,  ^.g.,  the  line  L^L^  when 
referred  to  the  axes  OX  and  OY,  has  the  equation 

«/  =  tan  6  -x  +  b, 

but  when  referred  to  the  axes  O'X'  and  0'  Y'^  the  former 
of  which  is  parallel  to  the  given  line,  its  equation  is  ?/  =  c. 

For  these,  and  other  reasons,  in  the  study  of  curves  and 
surfaces  by  the  methods  of  analytic  geometry,  it  will  often 
be  found  advantageous  to  transform  the  equations  from  one 
set  of  axes  to  another. 

It  will  be  found  that  the  coordinates  of  a  point  with 
reference  to  any  given  axes,  are  always  connected  by  simple 
formulas  with  the  coordinates  of  the  same  point  when  it  is 
referred  to  any  other  axes.  These  relations  or  formulas 
for  the  various  changes  of  axes  are  derived  in  the  next  few 
articles. 

I.    CARTESIAN   COORDINATES  ONLY 

71.  Change  of  origin,  new  axes  parallel  respectively  to  the 
original  axes.  Let  OX  and  Oy  be  the  original  axes,  0' X' 
and  O'Y'  the  new  axes,  and  let  the  coordinates  of  the  new 

origin  when  referred  to  the 
original  axes  be  h  and  h,  ^.e., 
0'  =  (A,  A:),  where  h  =  OA  and 
k  =  AO'.  Also  let  P,  any  point 
of  the  plane,  have  the  coordi- 
nates X  and  y  when  it  is  referred 
to  the  axes  OX  and  OF,  and  x' 
and  y'  when  it  is  referred  to  the  axes  0' X'  and  0'  Y' . 
Draw  MM'P  parallel  to  the  ?/-axis  ;  then 

0M=  OA  +  AM=  OA  +  O'M', 
and  similarly,  y  =  y'  +  k. 


Fig.  57 


[23] 


70-71.]  TRANSFORMATION  OF  COORDINATES  125 

which  are  the  equations  (or  formulas)  of  transformation 
from  any  given  axes  to  new  axes  which  are  respectively 
parallel  to  the  original  ones,  the  new  origin  being  the  point 
0'  =  (h^k^.  These  formulas,  moreover,  are  independent 
of  the  angle  between  the  axes. 

As  a  simple  illustration  of  the  usefulness  of  such  a  change 
of  axes,  suppose  the  equation 

x^  -  2hx  -\- 1/  -  2ky  =  a^  -  h?  -  k'^       .     .     (1) 

given,  in  which  x  and  y  are  coordinates  referred  to  the  axes 
OX  and  or. 

Now  let  F  =  (2;,  ?/)  be  any  point  on  the  locus  L^L  of  this 
equation,  and  let  (a;',  ?/')  be  the  coordinates  of  the  same 
point  P  when  it  is  referred  to  the  axes  O'X'  and  0'  Y'  ; 
then 

x—x'-\-h     and     y  —  y'  -{-k. 

Substituting  these  values  in  the  given  equation  for  the 
X  and  y  there  involved,  an  equation  in  x'  and  y'  is  obtained 
which  is  satisfied  by  the  coordinates  of  every  point  on  X^i-, 
z.e.,  it  is  the  equation  of  the  same  locus.  The  substitution 
gives : 

(^'  +  A)2  -2h(x'  -h  h^  +  iy'  +  ky  -2k(iy'-irk^  =  a?-¥-k\ 

which  reduces  to 

a   much   simpler   equation  than  (1),  but   representing   the 
same  locus,  merely  referred  to  other  axes. 

EXERCISES 

1.  AVhat  is  the  equation  for  the  locus  of  3  a:  —  2  ?/  =  6,  if  the  origin 
be  changed  to  the  point  (4,  3),  —  directions  of  axes  unchanged  ? 

2.  What  does  the  equation  x^  +  y"  —  ix  -  6 y  =  IS  become  if  the 
origin  be  changed  to  the  point  (2,  3),  — directions  of  axes  unchanged? 


126 


ANALYTIC  GEOMETRY 


[Ch.  VI. 


3.  What  does  the  equation  y'^  —  2x^  —  27/  +  6x  —  3  =  0  become  ^Yhen 
the  origin  is  removed  to  (f,  1),  —  directions  of  axes  unchanged? 

4.  Find  the  equation  for  the  straight  line  y  =  3x  -h  I  when  the  origin 
is  removed  to  the  point  (1,  4),  — directions  of  axes  unchanged. 

5.  Construct  appropriate  figures  for  exercises  1  and  4. 

72.  Transformation  from  one  system  of  rectangular  axes 
to  another  system,  also  rectangular,  and  having  the  same 
origin :  change  of  direction  of  axes. 

Let  OX  and  OY  he  sl  given  pair  of  rectangular  axes,  and 
let  OX'  and  OF'  be  a  second  pair,  with  Z.XOX'  =  ^,  the 

angle  through  which  the  first  pair 
of  axes  must  be  turned  to  come 
into  coincidence  with  the  second. 
Also  let  P,  any  point  in  the 
plane,  have  the  coordinates  x 
and  y  when  it  is  referred  to  the 
first  pair  of  axes,  and  x'  and  y' 
when  referred  to  the  second  pair.  The  problem  now  is  to 
express  x  and  y  in  terms  of  x\  y\  and  6.  Draw  the  or- 
dinates  MP,  M'P,  and  QM',  and  draw  M'B  parallel  to  the 
ir-axis;  then 

OM  =  OQ  +  QM=  OM'  cos  6  -  M'F  sin  6*, 

i.e.,  a?  =  a?' cos 6  -  2/' sine,  1 

•     •     •     [24] 
and  similarly,  y  =  oc'  sinQ  -]-  y  cos 0,  J 

which  are  the  required  formulas  of  transformation  from  one 
pair  of  rectangular  axes  to  another,  having  the  same  origin 
but  making  an  angle  6  with  the  first  pair. 

Note  1.  These  formulas  are  more  easily  obtained, — in  fact,  they  can 
be  read  directly  from  the  figure,  —  if  one  recalls  Art.  17,  and  considers 
that  the  projection  of  OP  equals  the  projection  of  OM'  +  the  projection 
of  M'P,  upon  OX  and  OF  in  turn. 

ISToTE  2.  The  reader  will  observe  that  a  combination  of  the  trans- 
formation of  Art.  71  with  that  of  Art.  72  will  transform  from  one  pah 
of  rectangular  axes  to  any  other  pair  of  rectangular  axes. 


72-73.] 


TRANSFORMATION  OF  COORDINATES 


127 


EXERCISES 

Turn  the  axes  through  an  angle  of  45°,  and  find  the  new  equations 
for  the  following  loci : 

1.   x''  +  y^  =  16',  2.   a;2-3/2  =  i6;  3.   ^  ^  ^^  _  1 . 

4.  17  a;^- 16  a:^  +  17  2/2  =  225. 

5.  If  the  axes  are  turned  through  the  angle  tan-i2,  what  does  the 
equation  4:xy  —  3x^  =  a^  become  ? 

73.  Transformation  from  rectangular  to  oblique  axes,  origin 
unchanged.  Let  OX  and  OYhe  a>  given  pair  of  rectangular 
axes,  let  OX'  and  OY'  be 
the  new  axes  making  an  an- 
gle ft)  with  each  other,  and 
let  the  angles  XOX'  and 
XOY'  be  denoted  by  d 
and  <^,  respectively.  Also 
let  P,  any  point  in  the 
plane,  have  the  coordinates 

X  and  y  when  referred  to  the  first  pair  of  axes,  and  x'  and  y' 
when  referred  to  the  second  pair. 

Draw  the  ordinates  MP,  M'  P,  and  QM',  also  draw  M'B 
parallel  to  the  rr-axis. 

Then  0M=  OQ-hQM=  OM'  cos6'  +  MP  sin  (90  -  <^); 
i.e.,  x  =  x'  cos  0  -\-  y'  cos  </>,  1  * 

and  similarly,        y  =  ^'  sin  0  -\-  y'  sin  (/>,  J 
which   are    the   required   formulas    of   transformation   from 
rectangular  to  oblique  axes  having  the  same  origin. 

If  ft)  =  90°,  and  consequently  cf)  =  90*^  +  6,  then  formulas 
[25]  reduce  to  [24],  and  Art.  73,  therefore,  includes  Art.  72 
as  a  special  case. 

By  first  solving  for  x'  and  y\  formulas  [25]  may  also  be 
employed  to   transform   from   oblique   to   rectangular   axes. 


Y 

/         / 

/                                /90°4<p 

X 

0 

X 

-^7 

Q       M 

/ 

Fig.  59. 

[25] 


*  See  Note  1,  Art.  72. 


128 


AIsALYTlC  GEOMETRY 


[Ch.  VI. 


EXERCISES 

1.  Given  the  equation  ^x'^  —  \Qy^  =  144  referred  to  rectangular  axes; 
what  does  this  equation  become  if  transformed  to  new  axes  such  that  the 
new  a:-axis  makes  the  angle  tan-i(-|),  and  the  new  ^-axis  the  angle 
tan-i(f),  ^7ith  the  old  a:-axis,  —  origin  unchanged  ? 

2.  If  the  old  and  new  x-axes  coincide,  and  the  new  axes  are  rectan- 
gular while  the  old  axes  are  inclined  at  an  angle  of  60°,  what  are  the 
equations  of  transformation  from  the  old  axes  to  the  new?  From  the 
new  axes  to  the  old?    Origin  unchanged  in  each  case. 

3.  If  the  first  two  of  the  three  sides  of  a  triangle  whose  equations  are 
2y-\-x-\-\  =  0,^y  —  X— \  =  0,  and  2  a:  +  3  ^  =  1,  are  chosen  as  new  axes, 
find  the  new  equations  of  the  sides. 


M       Q 
Fig.  60. 


74.   Transformation  from  one  set  of  oblique  axes  to  another, 

origin  unchanged.  Let  OX 
and  OY  he  a  given  pair  of 
axes,  OX'  and  OY'  the  new 
axes,  and  let  the  angles  XO  F, 
X'OY',  XOX\  and  XOY' 
be  denoted  by  &>,  (d\  0^  and  (/>, 
respectively.  Also  let  P,  any 
point  in  the  plane,  have  the 
coordinates  x  and  y  when  referred  to  the  first  pair  of  axes, 
and  x'  and  y'  when  referred  to  the  second  pair. 

Draw  M'P   parallel   to   OF',  MP   and  QM'  parallel  to 
OY,  and  M' R  parallel  to  OX. 
Then,  from  the  triangle  OQM', 

sin  6 
sm  o)  sin  (o 

and  from  the  triangle  RM'P, 


0$  =  ^'?ijli^J^  and(?M' = 


BM'  =  y'  "'"(-^  -  ")  and  RP  =  y' ?Hli- 
sin  ©  sin  o) 

But      0M=  OQ-  RM',  and  MP  =  QM'  +  RP ; 


73-75.]  TRANSFORMATION  OF  COORDINATES  129 


...  ,^-^^sm(ft)-6>)  ^  ^,sm(ft)-(^)^ 


sill  (0  sm  CO 


, 1  I  sin  0   ,     .sin  6 

and  y  =  x' h  y' ^- 


siii  ft)  sin  ft) 


.    .    .    [26] 


which  are  the  required  formulas  of  transformation  from  one 
pair  of  oblique  axes  to  another  having  the  same  origin. 

Note.  If  it  is  desired  to  change  the  origin,  and  also  the  direction  of 
the  axes,  the  necessary  formulas  may  be  obtained  by  combining  Art.  71 
with  Art.  72,  Art.  73,  or  Art.  74,  depending  upon  the  given  and  required 
axes. 

EXERCISES 

1.  Show,  by  specializing  some  of  the  angles  w,  to',  9,  and  ^  in  Art.  74, 
that  formulas  [26]  include  both  [25]  and  [24]  as  special  cases. 

2.  The  equation  of  a  certain  locus,  when  referred  to  a  pair  of  axes 
that  are  inclined  to  each  other  at  an  angle  of  60°,  is7  x^  —  2xy  +  4:y'^  =  5  ; 
what  will  this  equation  become  if  the  axes  are  each  turned  through  an 
angle  of  30°?  What  if  the  x-axis  is  turned  through  the  angle  —30° 
while  the  ^-axis  is  turned  through  +  30°  ? 

75.  The  degree  of  an  equation  in  Cartesian  coordinates  is 
not  changed  by  transformation  to  other  axes.  Every  formula 
of  transformation  obtained  ([23]  to  [26])  has  replaced  rr  aiid 
y,  respectively,  by  expressions  of  the  first  degree  in  the  new 
coordinates  a?',  y'.  Therefore  any  one  of  these  transforma- 
tions replaces  the  terms  containing  x  and  ^  by  expressions 
of  the  same  degree,  and  so  cannot  raise  the  degree  of  the 
given  equation.  Neither  can  any  one  of  these  transforma- 
tions lower  the  degree  of  the  given  equation ;  for  if  it  did, 

*  These  formulas  can  also  be  read  directly  from  Fig.  60  by  first  project- 
ing OM  and  then  the  broken  line  OM'PM  upon  a  line  perpendicular  to  OY ; 
and  afterwards  projecting  MP  and  also  the  broken  line  MOM'P  upon  a  per- 
pendicular to  OX.  The  results  being  equated  in  each  case,  and  divided  by 
sin  w,  give  [26]. 

TAN.   AN.   GEOM.  9 


130 


ANALYTIC  GEOMETRY 


[Ch.  VI. 


then  a  transformation  back  to  the  original  axes  (which  must 
give  again  the  original  equation)  would  raise  the  degree, 
which  has  just  been  shown  to  be  impossible ;  hence  all  these 
transformations  leave  the  degree  of  an  equation  unchanged. 


II.     POLAR  COORDINATES 


76.  Transformations  between   polar  and   rectangular   sys- 
tems.    (1)   Transfoi^mation  from    a   rectangular    to    a  polar 

system^  and  vice  versa^  the  origin  and 
X-axis  coinciding  res2:)ectively  with  the 
pole  and  the  initial  line.  Let  OX 
and  0 1^  be  a  given  set  of  rectangular 
axes,  and  let  OX  and  0  be  the  initial 
line  and  pole  for  the  system  of  polar 
coordinates.  Also  let  P,  any  point  in  the  plane,  have  the 
coordinates  x  and  y  when  referred  to  the  rectangular  axes, 
and  p  and  6  in  the  polar  system  (Fig.  61),  then 


Fig.  61. 


similarly, 


0M=  OP  cos  XOF; 
a?  =  P  cos  0 ; 
1/  =  P  sin  9. 


[27] 


These  are  the  required  formulas  of  transformation  when,  hut 
only  when,  the  rectangular  and  polar  axes  are  related  as 
above  described. 

Conversely,  from  formulas  [27],  or  directly  from  Fig.  61, 


p  =  Va?^  +  y^,  cos  6  = 


X 


^Q^  +  ?/2 


,  and  sin  6 


y 


Va;^  +  y'^ 


[28] 


which   are   the    required   formulas  of   transformation   from 
polar  to.  rectangular  axes,  under  the  above  conditions. 


75-76.] 


TRANSFORMATION  OF  COORDINATES 


131 


(2)   jSame  as  (1)  except  that  the  initial  line  OR  makes  an 
angle   cc  with  the  x-axis.      It   is  at 
once   evident  that  the  formulas  of 
transformation  for  this  case  are ; 

X  =  p  cos(^  +  «),  1 
and    y  =  p  sin  (^  +  a).  J 

The  converse    formulas   for  this 
case  are : 

p  =  -\/x^  -\-  y^ 

and    Q  =  cos"^ 


X 


Vx^  +  y^. 


—  a  =  sni 


-1 


y 


Vx^  +  y' 


-a.     [30] 


(3)  Transformation  from  any  Cartesian  system  to  any  polar 
system.  Transform  first  to  rectangular  axes  whose  origin  is 
the  proposed  pole ;  this  is  accomplished  by  Arts.  71  and  73. 
Then  by  formula  [27]  or  [29]  transform  from  the  rectangular 
Cartesian  to  the  polar  coordinates. 


EXERCISES 

Change  the  following  to  the  corresponding  polar  equations;  draw  a 
figure  showing  the  two  related  systems  of  axes  in  each  case.  Take  the  pole 
at  the  origin,  the  polar  axis  coincident  with  the  axis  of  x,  in  exercises  1  to  4. 


1.   x^  -{■  y^  =  a' 


3.   a;2  +  2/2  =  9  (x^  -  y^). 


2.   y^  —  X  +  2ay  =  0.  4.   y  =  x  tan  a. 

,5.   X  —  VSz/  +  2  =  0,  taking  pole  at  origin,  polar  axis  making  the 
angle  60°  with  the  x-axis. 

6.  x'^  —  y^  —  4:x  —  Qy  —  54  =  0,  taking  the  pole  at  the  point  (2,  -3), 
and  the  polar  axis  parallel  to  the  x-axis. 

Change  the  following  to  corresponding  rectangular  equations.     Take 
the  origin  at  the  pole  and  the  x-axis  coincident  with  the  polar  axis. 

7.  p  =  a.  9.   p2  sin  2  ^  =  10. 

8.  p2cos2^  =  a2.  10.   p2  =  a2sin2^. 
Suggestion.     In  Ex.  10  multiply  by  p2  and  substitute  2  sin  6  cos  0  for 

sin  2  0 ;  the  equation  then  becomes  p^  =  2  a2  p2  gin  0  cos  0. 

11.  p  =  k  cos  e.  12.   ^  =  3  tan-12.  13.  p^  cos  -  =  F. 


1B2  ANALYTIC  GEOMETRY  [Ch.  VI. 

EXAMPLES    ON    CHAPTER   VI 

1.  Find  the  equation  of  the  locus  of  2xy-7x-\-iy  =  0  referred  to 
parallel  axes  through  the  point  (-2,  |). 

2.  Transform  the' equation    x^  -  4xi/ +  4:if  -  Q x  +  I2y  =  0  to   new 
rectangular  axes  making  an  angle  tan-i  |  with  the  given  axes. 

3.  Transform    y'^  —  xy  —  dx  +  by  =  i)  to    parallel  axes  through  the 
point  (-5,  -5).     Draw  an  appropriate  figure. 

4.  Transform  the  equation  of  example  3  to  axes  bisecting  the  angles 
between  the  old  axes.     Trace  the  locus. 

5.  To  what   point  must  the  origin  be  moved  (the  new  axes  being 
parallel  to  the  old)  in  order  that  the  new  equation  of  the  locus 

2x2-  5a;^  _  3^2_2a:  +  13y  -  12  =  0 
shall  have  no  terms  of  first  degree  ? 

Solution.     Let  the  new  origin  be  (h,  k)  ;  then  x  -  x'  +  li,  y  =  y'  +  k, 
and  the  new  equation  is 

2(x'  +  hy-o(x'^h)i2j'+k)-3(y'^ky-2(x'  +  h)  +  ld(y'+k)-12  =  0, 
U.,         2x'^  -  5x'y'  -  3/2  +  (^h  -  bk -2)x'  -  (bh  +  Qk-  13)/ 

+  2/i2  -  5M- -  3)^2  _  2A  +  13^  -  12  =  0; 
but  it  is  required  that  the  coefficients  of  x'  and  y'  shall  be  0 ;  i.e.,  h  and 
k  are  to  be  determined  so  that 

^h-bk-    2  =  0, 
and  5h-\-6k  -13  =  0; 

hence  A  =  -V-  and  k  =  f. 

Therefore  the  new  origin  must  be  at  the  point  (^^,  f),  and  the  new 

equation  is 

2  x'2  -  5  x'y'  -  3  /2  _  8  =  0. 

6.  The  new  axes  being  parallel  to  the  old,  determine  the  new  origin 
so  that  the  new  equation  of  the  locus 

x'^  -  3  a-^  +  ?/2  +  10  a:  -  10  ?/  +  21  =  0 
shall  have  no  terms  of  first  degree. 

7.  Transform  the  equations  x  +  y  —  3  =  0  and  2a;  —  3?/  +  4  =  0  to 
parallel  axes  having  the  point  of  intersection  of  these  lines  as  origin. 

8.  Transform  the  equation  j  +  ^  =  1  to  new  rectangular  axes  through 
the  point  (2,  3),  and  making  the  angle  tan  "^  (  — |)  with  the  old  axes. 

9.  Through  what  angle  must  the  axes  be  turned  that  the  new  equa- 
tion of  the  line  6x  +  4?/  —  24  =  0  shall  have  no  y-term ?  Show  this 
geometrically,  from  a  figure. 


76.]  TBANSFORMATION   OF  COORDINATES  133 

10.  Through  what  angle  must  the  axes  be  turned  in  order  that  the 
new  equation  of  the  line  6  a:  +  4  y  =  24  shall  have  no  x-term  ?  Show 
analytically  (cf.  also  examples  8  and  9). 

Solution.  Let  6  be  the  required  angle ;  then  the  equations  of  trans- 
formation are 

X  =  x'  cos  6  —  y'  sin  6  and   y  =  x'  sin  0  +  y'  cos  0 ; 

and  the  new  equation  is 

(6  cos  ^  +  4  sin  0)x  -  (tj  sin  6  -  4:cos0)y  =  24: ; 
but  it  is  required  that  the  coefficient  of  x  be  0, 

6cos^  +  4sin  ^  =  0,  i.e.,  tan^  =  — f; 
whence  ^  =  tan-i(  — f), 

and  the  equation  becomes 

(6  sin  ^  -  4  cos  ^)  ^  +  24  =  0, 

which  reduces  to  — ^:y  +  24  =  0, 

V13 

i.e.,  to  5z/  +  12Vl3  =  0. 

11.  Through  what  angle  must  the  axes  be  turned  to  remove  the 
a:-term  from  the  equation  of  the  locus  Ax  -\-  By  +C  =  0"^  to  remove 
the  y-term  ? 

12.  Show  that  to  remove  the  xy-term.  from  the  equation  of  the  locus, 
2  x^  —  o  xy  —  S  y^  =  S  (cf.  Ex.  5),  the  axes  must  be  turned  through  the 
angle  6  =  67°  30',  i.e.,  so  that  tan 2^  =  —  1.     What  is  the  new  equation? 

13.  Through  v\^hat  angle  must  a  pair  of  rectangular  axes  be  turned 
that  the  new  x-axis  may  pass  through  the  point  (  —  2,  —  5)  ? 

14.  What  point  must  be  the  new  origin,  the  direction  of  axes  being 
unchanged,  in  order  that  the  new  equation  of  the  line  Ax  -{■  By  -f  C  =  0 
shall  have  no  constant  term? 

15.  To  what  point,  as  origin  of  a  pair  of  parallel  axes,  must  a  trans- 
formation of  axes  be  made  in  order  that  the  new  equation  of  the  locus, 
xy  —  y'^  —  X  -\-  y  =  0,  shall  have  no  terms  of  first  degree?  Construct  the 
locus. 

16.  Find  the  new  origin,  the  direction  of  axes  remaining  unchanged, 
so  that  the  equation  of  the  locus,  x'^  +  xy  —  'd  x  —  y  -\-  2  =  0,  shall  have 
no  constant  term.     Construct  the  figure. 

17.  Transform  the  equation  4^2  -f  2V3xy  +  2y^  =  1  to  new  rectan- 
gular axes  making  an  angle  of  30°  with  the  given  axes,  —  origin  unchanged. 


134  ANALYTIC  GEOMETRY  [Ch.  VL  76. 

18.  Transform  y^  =  Sx  to  new  rectangular  axes  having  the  point 
(18,  12)  as  origin,  and  making  an  angle  cot-i3  with  the  old. 

19.  Transform  to  rectangular  coordinates,  the  pole  and  initial  line 
being  coincident  with  the  origin  and  x-axis,  respectively : 

(a)    p^  =  a^  cos  2  6,      (/3)    p2cos2(9  =  a2,      (y)    p  =  ^-sin2^. 

Transform  to  polar  coordinates,  the  ar-axis  and  initial  line  being  coin- 
cident : 

20.  (x^  +  y^y  =  k^(x^  -  ?/2),  the  pole  being  at  the  point  (0,  0)  ; 

21.  a;2  +2/2  —  7  ^^^  pole  being  at  the  point  (0,  0)  ; 

22.  x^  +  y^  =  IQx,  the  pole  being  at  the  point  (8,  0). 

23.  Transform  the  equation  y^  +  4:ay  cot  30°  —  4  ax  =  0  to  an  oblique 
system  of  coordinates,  with  the  same  origin  and  a:-axis,  but  the  new 
y-axis  at  an  angle  of  30°  with  the  old  x-axis. 

24.  Transform   the   equation   t^  +  ^  =  1>  to  new  axes,  making  the 

positive  angles  tan  -i  |  and  tan  ~^(—  f),  respectively,  with  the  old  a;-axis, 
the  origin  being  unchanged. 

25.  Transform  the  equation 

3  x2  +  10V3  xy  -  7  2/2  =  (18  -  SOVS)  x  +  (42  +  30\/3)  ?/  +  (42  +  90\/3) 

to  the  new  origin  (3,  —3),  with  new  axes  making  an  angle  of  30°  with 
the  old. 

26.  Transform  the  equation  3  a;^  +  8  x?/  —  3  2/^  =  0  to  the  two  straight 
lines  which  it  represents,  as  new  axes. 

27.  Transform ^  =  1  to   the  straight  lines —  =  0,  as  new 

25      9  ^  25       9 

axes. 

28.  Transform  to  polar  coordinates  the  equation  y^  (2  a  —  x)  =  x^. 

29.  Transform  to  rectangular  coordinates  the  equation 

p  i=a(cos2^+ sin2(9). 

30.  Prove  the  formula  for  the  distance  in  polar  coordinates  [1]  by 
transformation  of  the  corresponding  formula  [2]  in  rectangular  coordi- 
nates. 

31.  Transform  the  equation  x  cos  a  +  ?/  sin  a=  p  io  polar  coordinates. 


CHAPTER   VII 
THE  CIRCLE 
«    Special  Equation  of  the  Second  Degree 

77.  It  must  be  kept  clearly  in  mind  that  one  of  the  chief 
aims  of  an  elementary  course  in  Analytic  Geometry  is  to 
teach  a  new  method  for  the  study  of  geometric  properties  of 
curves  and  surfaces.  Power  and  facility  in  the  use  of  such 
a  new  method  are  best  acquired  by  applying  it  first  to  those 
loci  whose  properties  are  already  best  understood.  Accord- 
ingly, the  straight  line  having  already  been  studied  in 
Chapter  V,  the  circle  will  next  be  examined. 

It  will  appear  later  that  the  circle  is  only  a  special  case  of 
the  conic  sections  already  referred  to  in  Art.  48,  and  might, 
therefore,  be  advantageously  studied  after  the  general  prop- 
erties of  those  curves  had  been  examined  ;  the  present  order 
is  adopted,  however,  because  the  student  is  already  familiar 
with  the  chief  properties  of  the  circle. 

In  solving  the  exercises  of  this  chapter  the  student  should 
use  the  analytic  methods,  even  when  purely  geometric  methods 
might  sufhce,  —  he  is  learning  to  use  a  new  instrument  of 
investigation,  and  is  not  merely  studying  the  properties  of 
the  circle. 

78.  The  circle:  its  definition,  and  equation.  The  circle  may 
be  defined  as  the  path  traced  by  a  point  which  moves  in  such 
a  way  as  to  be  always  at  a  constant  distance  from  a  given 
fixed  point.  This  fixed  point  is  the  center,  and  the  constant 
distance  is  the  radius,  of  the  circle. 

135 


136  ANALYTIC   GEOMETRY  [Ch.  VII. 

To  derive  the  equation  from  this  definition,  let  (7=  (A,  k') 
be  the  center,  r  the  radius,  and  JP  =  (a;,  z/)  any  point  on  the 

->^  curve.     Also  draw  the  ordinates 

^      M^O  and  MP,  and  the  line  OE 
i  I      parallel  to  the  a;-axis  ;   then 

I  /       OP  =  r  ;     [geometric    equation] 

>/        but  (Art.  26), 


M-, — M^      OP  =  V(^x  -  hy  +  (z/  -  ky, 

hence    V(a;  —  A)^  +  (^  ~  ■^)^  =  ^'  5 
i.e.,  (x-7i)^  +  (y-k)^  =  r^,*       .        .        .        [31] 

which  is  the  equation  of  the  circle  whose  radius  is  r,  and 
whose  center  has  the  coordinates  h  and  k. 

With  given  fixed  axes,  equation  [31]  may,  by  rightly 
choosing  A,  k^  and  7%  represent  any  circle  Avhatever  ;  it  is, 
therefore,  called  the  general  equation  of  the  circle.  Of  its 
special  forms  one  is,  because  of  its  very  frequent  applica- 
tion, particularly  important ;  this  form  is  the  one  for  which 
the  center  coincides  with  the  origin  :  in  that  case  h  =  k  =  0, 
and  equation  [31]  becomes 

^2  +  2/2  =  ^,2.1  .  .  .  [32] 

*  Equation  [31]  may  be  written  in  the  form 

(X  -  hy  +  iy-  ky  -  r2  =  0  ; 
the  first  member  then  becomes  positive  if  the  coordinates  of  any  point  outside 
of  the  circle  are  substituted  for  x  and  y,  it  becomes  negative  for  a  point  inside 
of  the  circle,  and  zero  for  a  point  on  the  circle.  Hence  the  circle  may  be 
regarded  as  the  boundary  which  separates  that  part  of  the  plane  for  which 
the  function  (x  —  h)'^  +  (^  —  A;)^  —  r^  is  positive  from  the  part  for  which  this 
function  is  negative.  The  inside  of  the  circle  may  therefore  be  called  its  nega- 
tive side,  while  the  outside  is  its  jiositive  side  (cf.  foot-note,  Art.  4.3). 

t  If  one  is  unrestricted  in  his  choice  of  axes,  then  an  equation  of  the  form 
of  [32]  may  represent  any  circle  whatever,  —  the  axes  need  merely  be  chosen 
perpendicular  to  each  other  and  through  its  center;  —  equation  [31]  is  more 
general  in  that,  the  rectangular  axes  being  determined  by  other  considera- 
tions, it  may  still  represent  any  circle  whatever. 


78-79.]  THE  CIRCLE  137 

EXERCISES 
First  construct  the  circle,  then  find  its  equation,  being  given 

1.  the  center  (5,  -3),  the  radius  4; 

2.  the  center  (0,  2),  the  radius  |; 

3.  the  center  (3,  -3),  the  radius  3 ; 

4.  the  center  (0,  0),  the  radius  5; 

5.  the  center  (-4,  0),  the  radius  1. 

6.  How  are  circles  related  for  which  h  and  k  are  the  same,  while  ?-  is 
different  for  each  ?  for  which  h  and  r  are  the  same,  while  k  differs  for 
each  ? 

7.  What  form  does  the  equation  of  the  circle  assume  when  the  center 
is  on  the  a:-axis  and  the  origin  on  the  circumference?  when  the  circle 
touches  each  axis  and  has  its  center  in  quadrant  11? 

79.   In  rectangular  coordinates  every  equation  of  the  form 

cc  -+  2/'^  +  2  Goc  +  2  i^'i/  +  C  =  0  represents  a  circle.  The  equa- 
tions of  the  circles  already  obtained  (equations  [31]  and 
[32],  as  well  as  the  answers  to  examples  1  to  5  and  7)  are  all 
of  the  form 

x^^-y'^  +  2ax+2Fy-{-C=0;    .     .     .     (1) 

it  will  now  be  shown  that,  for  all  values  of  ^,  F^  and  (7, 
for  which  the  locus  of  equation  (1)  is  real,  this  equation 
represents  a  circle. 

To  prove  this  it  is  only  necessary  to  complete  the  square 
in  the  ^-terms  and  in  the  ?/-terms,  by  adding  G^  -f  F^  to  each 
member  of  equation  (1),  and  then  transpose  C  to  the  second 
member.     Equation  (1)  may  then  be  written  in  the  form 

(x  +  G-y  +  (?/  +  Ff  =  a'^  +  F^-  0 

=  (V(/2  +  i^2_    (7)2      .        ,        (2) 

which  is  (cf.  equation  [31])  the  equation  of  a  circle  Avhose 
center  is  the  point  (—  (r^,  —  ^),  and  whose  radius  is 


138  ANALYTIC  GEOMETRY  [Ch.  VII. 

Note  1.     This  circle  is  real  only  ii  G^  +  F^  -  C>0;  for,  if 

G^+F-2-  C<0, 
its  square  root  is  imaginary,  and  no  real  values  of  x  and  ,?/can  then  satisfy  __ 
equation  (2)  ;  while  if  6'"^  +  F^  -  C  =  0,  then  equation  (2)  reduces  to 

(x+Gy  +  (y  +  Fy  =  0, (3) 

which  may  be  called  the  equation  of  a  "  point  circle,"  since  the  coordi- 
nates of  but  one  real  point,  viz.  {-G,  -F),  will  satisfy  equation  (3). 
If,  however,  G'^  +  F'^  -C>0,  then  equation  (1)  represents  a  real  circle 
for  all  values  of  G,  F,  and  C,  subject  to  this  single  limitation. 

Note  2.  Every  equation  of  the  form  Ax'^  -i-Ay^ +  2Gx  -\- 2  Fij +  C  =  Q 
represents  a  circle,  for,  by  Art.  38,  this  equation  has  the  same  locus  as 

C  F  G 

has  a:^+  y^  -\-  2—  x  -\-  2  —  y  -^  —  =  0,  and   this   last  equation  is  of   the 
A  A  A 

form  of  equation  (1). 

Hence,  interpreted  in  rectangular  coordinates,  every  equation 
of  the  second  degree  from  which  the  term  hi  xy  is  absent,  and 
in  tvhich  the  coefficient  of  x^  equals  that  of  /,  represents  a 
circle. 

80.  Equation  of  a  circle  through  three  given  points.  By 
means  of  equation  [31],  or  of  the  equation 

^2  +  ^2  +  2fe  +  2^^/+  6^=0,   .     .     .     (1) 

which  has  been  shown  in  Art.  79  to  be  its  equivalent, 
the  problem  of  finding  the  equation  of  a  circle  which  shall 
pass  through  any  three  given  points  not  lying  on  a  straight 
line  can  be  solved ;  i.e.,  the  constants  h,  k,  and  r,  or  (r,  F, 
and  Q,  may  be  so  determined  that  the  circle  shall  pass 
through  the  three  given  points. 

To  illustrate  :  lot  the  given  points  be  (1,  1),  (2,  "l),  and 
(3,  2),  and  let  x^  +  y"^  ^- "IGx  +  2Fy  ■\- C  =  Q  be  the  equa- 
tion of  the  circle  that  passes  through  these  points ;  to  find 
the  values  of  the  constants  Gr,  F,  and  C.  Since  the  point 
(1,  1)  is  on  this  circle,  therefore  (cf.  Art.  35), 


79-80.]  -  THE  CIRCLE  139 

similarly,  4  +  l+4(7-2i'  +  (7  =  0, 

and  9  +  4  +  6(74-41^  +  (7  =  0. 

These  equations  give:  6^  =  — |-,  F  ~ —^,  and  (7=4. 
Substituting  these  values,  the  equation  of  the  required 
circle  becomes 

its  center  is  at  the  point  (f,  ^),  while  its  radius  is  ^VTO- 

Note.  The  fact  that  the  most  general  equation  of  the  circle  contains 
three  parameters  (Ji,  k,  and  r,  or  G,  F,  and  C,  above)  corresponds  to  the 
property  that  a  circle  is  uniquely  determined  by  three  of  its  points. 

EXERCISES 

Find  the  radii,  and  the  coordinates  of  the  centers,  of  the  following 
circles ;  also,  draw  the  circles. 

1.  x2  +  2/2  -  4  a:  -  8  2/  -  41  =  0.  ^.   2  (x^-  +  i/)  =  7  y. 

2.  3 x'^  +  3 y^  —  5 X  —7 y  -\- 1  =  0.  5.    ax^  +  ay'^  =  bx  -\-  cy. 

3.  x^  +  y^  =  3  (x  +  3).  6.    (x  +  yy  +  (x  -  y)'^  =  8  a^. 

7.  What  loci  are  represented  by  the  equations 

(x  -  hy  +  {y  -  ky  =  0, 
and  x2  +  ?/2-2a;  +  6?/  +  38  =  0? 

Find  the  equation  of  the  circle  through  the  points : 

8.  (1,2),  (3, -4),  and  (5,-6); 

9.  (0,  0),  (a,  b),  and  (b,  a)  ; 

10.  (-6,  -1),  (0,  1),  and  (1,  0)  ; 

11.  (10,  2),  (3,  3),  and  having  the  radius  2. 

12.  Find  the  equation  of  the  circle  which  has  the  line  joining  the 
points  (3,  4)  and  ("1,  2)  for  a  diameter. 

13.  Find  the  equation  of  the  circle  which  touches  each  axis,  and 
passes  through  the  point  (~2,  3). 

14.  A  circle  has  its  center  on  the  line  3x  -\-  iy  =  7,  and  touches  the 
two  lines  x  +  y  =  3  and  x  —  y  =  3]  find  its  equation,  radius,  and  center ; 
also  draw  the  circle. 


140 


ANALYTIC   GEOMETRY 


[Ch.  VII. 


SECANTS,    TANGENTS,    AND   NORMALS 

81.  Definitions  of  secants,  tangents,  and  normals.  A  straight 
line  will,  in  general,  intersect  any  given  curve  in  two  or  more 

distinct  points  ;  it  is  then  called  a 
secant  line  to  the  curve.  Let  P^ 
and  Pg  ^6  two  successive  points  of 
intersection  of  a  secant  line  P^P^Q 


Fig.  64. 


with  a  given  curve  LP^P^  ...  K; 
if  this  secant  line  be  rotated  about 
the  point  P^  so  that  P^  approaches 
Pj  along  the  curve,  the  limiting 
position  P^T  which  the  secant  approaches,  as  P^  approaches 
coincidence  with  P^,  is  called  a  tangent  to  the  curve  at  that 
point.  This  conception  of  the  tangent  leads  to  a  method,  of 
extensive  ap^Dlication,  for  deriving  its  equation,  —  the  so- 
called  "  secant  method."  * 

Since  the  points  of  intersection  of  a  line  and  a  curve  are 
found  (Art.  39)  b}^  considering  their  equations  as  simulta- 
neous, and  solving  for  x  and  ^,  it  follows  that,  if  the  line  is 
tangent  to  the  curve,  the  abscissas  of  two  points  of  intersec- 
tion, as  well  as  their  ordinates,  are  equal.  Therefore,  if  the 
line  is  a  tangent,  the  equation  obtained  by  eliminating  x  or 
y  between  the  equation  of  the  line  and  that  of  the  curve 
must  have  a  pair  of  equal  roots. 

If  the  given  curve  is  of  the  second  degree,  then  the  equa- 
tion resulting  from  this  elimination  is  of  the  second  degree, 
and  the  test  for  equal  roots  is  well  known  (Art.  9)  ;  but  if 
the  given  equation  is  of  a  degree  higher  than  the  second, 
other  methods  must  in  general  be  used. 

A  straight  line  drawn  perpendicular    to    a   tangent   and 


*  For  illustration,  see  Art.  84. 


81-82.] 


THE  CIRCLE 


141 


through  the  point  of  tangency  is  called  a  normal  line  to  the 
curve  at  that  point.  Thus,  in  Fig.  64,  PiP^)  ^i^z  ^^^  se- 
cants, PjT  is  a  tangent,  and  P^N  d  normal  to  the  curve  at  P^ 

82.   Tangents  :  Illustrative  examples. 

(1)  To  find  the  equation  of  that  tangent  to  the  circle  x^  -\-  y^  =  b 
which  makes  an  angle  of  45°  with  the  a:-axis.  Since  this  line  makes  an 
angle  of  45°  with  the  a:-axis  its  equation  is  y  =  x  -\-  h,  where  h  is  to  be 
determined  so  that  this  line  shall  touch  the  circle. 

Clearly,  from  the  figure,  there  are  two  values  of  h  (OB^  and  OB2)  for 
which  this  line  will  be  tangent  to  the 
circle.  According  to  Art.  81,  these 
values  of  b  are  those  which  make  the  two 
points  of  intersection  of  the  line  and  the 
circle  become  coincident. 

Considering  the  equations  x^  +  ^2  _  5 
and  y  =  X  +  b  simultaneous,  and  elimi- 
nating y,  the  resulting  equation  in  x  is 

x^+(x  +  by  =  o,  i.e.,  2x^  +  2bx-\-b^-5  =  0. 
The  roots  of  this  equation  will  become 
equal,  i.e.,  the  abscissas  of  the  points  of 
intersection  will  become  equal  (Art.  9), 
if  62  -  2  (62  _  5)  =  0,  i.e.,  iib  =  ±  VlO. 

The  equations  of  the  two  required  tangent  lines  are,  therefore, 
y  =  x  +  VlO,   and   y  =  x—VlO. 

(2)  To  find  the  equations  of  those  tangents  to  the  circle  x^  +  y^  =  Q  y 
that  are  parallel  to  the  line  x+2y  +  ll=0. 

The  equation  of  a  line  parallel  tox  +  2y  +  ll  =  0isx  +  2y-]-k  =  0, 
where  k  is  an  arbitrary  constant  (Art.  62),  and  this  line  will  become 
tangent  to  the  circle,  if  the  value  of  the  constant  k  be  so  chosen  that  the 
two  points  in  which  the  line  meets  the  circle  shall  become  coincident. 

Considering  the  equations  x^  -\-  y'^  =  Qy  and  x-\-2y-\-k  =  0  simulta- 
neous, and  eliminating  x,  the  resulting  equation  in  y  is 

{-  k  -2yy  +  y^=Qy,  i.e.,  5 y^  +  (4:  k  -  Q) y  +  k^  =  0. 
The  two  values  of  y  will  become  equal  if  (Art.  9) 

(4:k  -  6)2-20  ^•2  =  0,  i.e.,  if  t^  +  15  y^  -  9  =  0, 
i.e.,  if  k  =  —  6  ±3  Vo, 

and  the  two  required  tangent  lines  are : 

X  -\-2y  -e  +  3V5  =0,    and   x  +  2y  -  6  -  SVd  =  0. 


142  ANALYTIC  GEOMETRY  [Ch.  VII. 

EXERCISES 
Find  the  equations  of  the  tangents  : 

1.  to  the  circle  x'^  -f  ?/2  _  4^  parallel  to  the  line  a:  +  2y4-3  =  0; 

2.  to  the  circle  3(x2  +  2/"-^)  =^y<,  perpendicular  to  the  line  x'  +  ?/  =  7; 

3.  to   the   circle   x^  +  y'^  -^  \0  x  —  Q  y  —  2  =  {),    parallel  to   the    line 
y  =  '2x-1; 

4.  to  the  circle  x^  +  ?/2  —  ^2^  ^j^d  forming  with  the  axes  a  triangle 
whose  area  is  r^. 

5.  Show  that  the  line  y  =  x  -\-  cV2  is,  for  all  values  of  c,  tangent  to 
circle  x'^  +  ?/2  =  c^;  and  find,  in  terms  of  c,  the  point  of  contact. 

6.  Prove  that   the   circle   x'^  +  y^-\-2x-\-2y-\-l  =  {)  touches   both 
coordinate  axes ;  and  find  the  points  of  contact. 

7.  For  what  values  of   c  will  the   line  3x  —  4y  +  c  =  0  touch  the 
circle  x2  +  ?/2  _  8.r  +  12?/ -  44  =  0? 

8.  For  what  value  of  r  will  the  cu'cle  x'^  -\-  y^  =  r^  touch   the   line 
J/  =  3  X  —  5  ? 

9.  Prove  that   the   line  ax  =  b  (y  —  b)  touches  the  circle  x  (x  —  a) 
+  y  (y  —  b)  =  0 ;  and  find  the  point  of  contact. 

10.  Three  tangents  are  drawn  to  the  circle  x'^  +  y^  =  9]  one  of  them 
is  parallel  to  the  x-axis,  and  together  they  form  an  equilateral  triangle. 
Find  their  equations,  and  the  area  of  the  triangle. 

83.  Equation  of  tangent  to  the  circle  x^  +  y^  =  r^  in  terms 
of  its  slope.  The  equation  of  the  tangent  to  a  given  circle, 
in  terms  of  its  slope,  is  found  in  precisely  the  same  way  as 
that  followed  in  solving  (1)  of  Art.  82.  Let  m-^  be  the 
given  slope  of  the  tangent,  then  the  equation  of  the  tangent 
is  of  the  form 

7/  =  m^x  +  b,         .  .  .  (1) 

wherein  J  is  a  constant  which  must  be  so  determined  that 
line  (1)  shall  intersect  the  circle 

2;2  +  i/2  =  r2  .  .  .  (2) 

in  two  coincident  points. 


82-83.]  THE  CIRCLE  143 

Eliminating  y  between  equations  (1)  and  (2)  gives 

x^  +  (m^x  +  5)2  =  r^, 
^.6.,  x^  (1  +  m-^^)  -\-  2  hm-^x  +  5^  _  ^2  _  q  . 

and  the  two  values  of   x  obtained  from  this  equation  will 
become  equal  (Art.   9)  if 

Qm^hy  -  (1  +  ^^2)  (^2  _  ^2)  ^  0^ 


z.e.,  if  6  =  ±  r  Vl  +  ^1^- 

Substituting  this  value  of  h  in  equation  (1),  it  becomes 

y  =  miX  ±r^/l  +  rni^*      .       .       .       [33] 

which  is  then,  for  all  values  of  mi,  tangent  to  the  circle  (2). 
This  equation  [33]  enables  one  to  write  down  immediately 
the  equation  of  a  tangent,  of  given  slope,  to  a  circle  whose 
center^  is  at  the  origin. 

E.g.,  to  find  the  equation  of  the  tangent  whose  slope  ?nj  =  1  =  tan  45°, 
to  the  circle  x^  +  y^  =  5,  it  is  only  necessary  to  substitute  1  for  m^  and 
Vo  for  r  in  equation  [33].  This  gives  as  the  required  equation 
ij  =  x±  ViO  [of.  (1)  Art.  82]. 

Note  1.  If  the  center  of  the  given  circle  is  not  at  the  origin,  i.e., 
if  its  equation  is  of  the  form  x^  +  y^ +  2  Gx  -{-2  Fy  +  C  =  0,  instead  of 
x^+y^  =  r%  then  the  same  reasoning  as  that  employed  above  would  lead  to 

y  +  F  =  m^(x+G)±VG^  +  F^-C-Vl-\-m^^  .      .    [34] 
as  the  equation  of  the  required  tangent. 

This  equation  might  have  been  obtained  also  by  first  transforming 
the  equation  x"^  +  y'^  +  2  Gx  -\-  2  Fy  -\-  C  =  0  to  parallel  axes  through  the 
point  (-G,  -F)  ;  this  would  have  given  x'^  -i-  y'^  =  G^  +  F'^  -  C  =  r^ 
as  the  equation  of  the  sayne  circle,  but  now  referred  to  axes  through  its 
center.  Referred  to  these  new  axes  y'  =  m^x' ±r  Vl  +  m^^  (see  eq.  [33]) 
is,  for  all  values  of  w^  tangent  to  this  circle;  transforming  this  last 
equation  back  to  the  original  axes,  i.e.,  substituting  for  x',  y',  and  r  their 
equals,  viz.,  x  -\-  G,  y  -\-  F,  and  V^r^  ^  p-i  _  (j^  \^  becomes 

y  +  F=m^{x+  G)  ±  V^^  ^F'^-C  .  vTTw^ 
*  This  equation  is  sometimes  spoken  of  as  the  magical  equation  of  the  tangent. 


144 


ANALYTIC  GEOMETBT 


[Ch.  VII. 


as  before    which  is,  for  all  values  of  7n^,  tangent  to  the  circle  whose 
center  is  at  the  point  (-G,  "F)  and  whose  radius  is  \/G'^  +  F'^-  C. 

Note  2.  Because  of  its  frequent  occurrence,  it  is  useful  to  memorize 
equation  [33]-  On  the  other  hand,  it  is  not  recommended  that  equation 
[34]  be  memorized,  but  it  should  be  carefully  worked  out  by  the  student. 
Instead  of  employing  either  of  these  formulas,  however,  the  student 
may  always  attack  the  problems  directly,  as  was  done  in  Art.  82, 

EXERCISES 

Find  the  equations  of  the  lines  which  are  tangent : 

1.  to  the  circle  x^  +  y-  =  16,  and  whose  slope  is  3 ; 

2.  to  the  circle  x-  +  y-  =  4,  and  which  are  parallel  to  the  line  x  +  2  y 
+  3=0  (cf.  Ex.  1,  Art.  82); 

3.  to  the  circle  x^  -{-  y^  =  9,  and  which  make  an  angle  of  60*^  with  the 
a;-axis ;  with  the  ?/-axis ; 

4.  to  the  circle  x'^  +  y'^  =  25,  and  which  are  perpendicular  to  the  line 
joining  the  points  ("3,  7)  and  (7,  5)  ; 

5.  to  the  circle  x^  -^  y-  =  2  x  -\-  2  y  —  1,  and  whose  slojDe  is  ~1. 

84.  Equation  of  tangent  to  the  circle  in  terms  of  the  coordi- 
nates of  the  point  of  contact :  the  secant  method. 

(a)  Center  of  the  circle  at  the  origin.  Let  jP^  =  (i^i,  i/i)  be 
the  point  of  tangency,  on  the  given  circle 

x^  +  y^  =  r^.  .  .  .         (1) 

Through  Pi  draw  a  secant  line  LM,  and  let  Pg  =  (2:2,  ^2) 
be  its  other  point  of  intersection  with  the  circle.     If  the 

point  -P2  raoves  along  the  circle 
until  it  comes  into  coincidence 
with  _Pi,  the  limiting  position  of 
the  secant  LM  is  the  tangent 
PiT.     (Art.  81.) 

The  equation  of  the  line  L3I  is 

y-y^^'hUVl^X-Xi).   ...    (2) 
2^2  —  .^1 

Fig.  66.  If  now  P2  approaches  Pj  until 


83-84.]  THE  CIRCLE  145 

X2=Xi  and  y2=yn  equation  (2)  takes  the  indeterminate  form 

y-yi=-^(p^-^i)'      •      •      •       (3) 

This  indeterminateness  arises  because  account  has  not  yet 
been  taken  of  the  path  (or  direction)  by  which  P2  shall 
approach  Pi,  and  it  disappears  immediately  if  the  condition 
that  P2  is  to  approach  Pi  along  the  circle  (1)  is  introduced. 

Since  the  fixed  point  P^  is  on  the  circle  (1),  therefore 

^i'  +  ^i'=^';      ...      (4) 

and  since  P25  while  approaching  P^,  always  remains  on  circle 
(1),  therefore 

x}  ^  y^  =  T^\         .         ,         .         (5) 

hence,  subtracting  equation  (4)  from  equation  (5), 

^2  -  ^i  +  y2  -  Vi  =  0, 
that  is,        (?/2  -  y{)  (^2 + yO  =  -  (^2  -  ^1)  (^2 + ^1)  ; 

whence,  ^ — ^  = ^    ^^ . 

^2-^1  ^2  +  ^1 

Substituting  this  result  in  equation  (2)  gives 

y-y^^-'^Q.-X,^,^         ...         (6) 

which  is  the  equation  of  the  secant  line  LM  of  the  given 
circle  (1). 

*  The  difference  between  equations  (2)  and  (6)  consists  in  this  :  no  mat- 
ter where  the  points  (xi,  yi)  and  (X2,  2/2)  niay  be,  equation  (2)  represents 
the  straiglit  line  passing  through  them  ;  but  equation  (6)  is  the  equation  of 
the  line  through  these  points  only  when  they  are  on  the  circle  orP-  +  y'^  =  r^. 
In  other  words,  equation  (2)  is  the  equation  of  the  line  passing  through  any 
two  points  whatever,  while  equation  (6)  is  the  equation  of  the  line  passing 
through  any  two  points  on  the  circumference  of  the  circle. 

TAN.    AN.    GEOM.  —  10 


146  ANALYTIC   GEOMETRY  [Ch.  VIL 

Now  let  P2  mo^'e  along  the  circle  until  it  coincides  with 
Pi,  i.e.,  until  2^2  =  ^1'  ^^^^  ^2  =  ^/11  then  equation  (6)  becomes 

i.e.,  y-yi=  -  —(x-x{), 

yi 

which,  by  clearing  of  fractions  and  transposing,  may  be 
written  in  the  form 

xi^+yiy=^i-^yi^ 

i.e.,  i»ia?  +  2/12/ =  r2,        .         .         .         [35] 

which  is  the  required  equation  of  the  tangent  to  the  circle 
a;2  _|_  ^2  _  y.2^  ^^  ^^^  y^  being  the  coordinates  of  the  point  of 
tangency. 

(y8)    Center  of  circle  not  at  origin.     If  the  equation  of  the 
given  circle  be 

x^  +  i/+2ax  +  2Fy  +  0=^,   ...     (7) 

then,  Pj  and  P^  being  on  this  circle, 

^^2  +  ^^2+26^a:i  +  2P^i  +  C=0,   ...      (8) 
and  x^^+y^^  +  ^ax^^  +  'lFy^  +  C^O.   .     .     .     (9) 

Subtracting  equation  (8)  from  equation  (9), 
x,^  -  x{-  +  2  (^(rr^  -  2:1)  +  y.^  -  ^1^  +  2  P(^2  -  ^1)  =  0, 
which  may  be  written  in  the  form 

(^2  -  ^i)(^2  +  ^1  +  2  P)  =  -  (a;2  -  x{)(x^  +  a^i  4-  2(7); 


whence, 


y^- y\ _    x^-\-x^-\-2G 


^2-^1       2/2  +  2/1  +  2^ 

Substituting  this  result  in  equation  (2)  gives 

_x^±^h±l^r^_^^     .    .     .    (10) 
'  ^2  +  ^i  +  2P^  1^'  ^     ^ 


84-85.]  THE  CIRCLE  147 

as  the  equation  of  the  secant  through  the  two  points  (x^^  y-^) 
and  (a^g,  ^2)  ^^  '^^  circle  (7).  If,  now,  the  point  {x^^  y^ 
moves  along  the  curve  until  it  comes  into  coincidence  with 
(j^v  Vi)^  ^^^^  secant  line  becomes  a  tangent,  and  its  equation  is 

y-^i  =  -^^_^^(^-^i)-       •       •       •       (11) 

Clearing  equation  (11)  of  fractions,  and  transposing,  it 
may  be  written  thus  : 

x^x  +  y^y +  ax  + Fy  =  x^  +  y^  +  ax^  +  Fy^:   .   .   .   (12) 

but,  by  equation  (8),  the  second  member  of  equation  (12) 

equals 

—  G-x^  —  Fy^  —  C. 

Putting  this  value  for  the  second  member  in  equation  (12), 
and  transposing,  that  equation  becomes 

xioc^-viv +G(dc  +  Xi)+F{y +  yi) +  C  =  0,  .    .    .    [36] 

which  is  the  required  equation  of  the  tangent  to  the  circle 
(7),  x-^  and  y^  being  the  coordinates  of  the  point  of  contact.* 

XoTE.  Equation  [36]  may  be  easily  remembered  if  it  be  observed 
that  it  differs  from  the  equation  of  the  circle  [equation  (7)]  only  in 
having  x^x,  y-^ij,  x  +  x-^,  and  y  +  y^io.  place  of  x^,  y^,  2  a;,  and  2y,  respec- 
tively. Tt  will  be  found  later  that  any  equation  of  the  second  degree 
(from  which  the  a;y-term  is  absent)  bears  this  same  relation  to  the  equa- 
tion of  a  tangent  to  its  locus,  x^  and  y^  being  the  coordinates  of  the  point 
of  contact.     Compare,  also,  equation  [35]  with  equation  (1). 

It  must  also  be  carefully  kept  in  mind  that  equations  [35]  and  [36] 
represent  tangents  only  if  (a'^,  ?/j)  is  a  point  on  the  circle.  It  will  be  seen 
later  that  these  equations  represent  other  lines  if  (a-^,  y^)  is  not  on  the  circle. 

85.  Equation  of  a  normal  to  a  given  circle.  By  definition 
(Art.  81)  the  normal  at  a  given  point,  P-^  =  (x-^^  y^),  on  any 

*  Equations  (11)  and  (12)  are,  of  course,  but  different  forms  of  the  equa- 
tion of  the  same  tangent  as  that  represented  by  equation  [36]. 


148  ANALYTIC  GEOMETRY  [Ch.  VII. 

curve  is  the  line  through  P^,  and  perpendicular  to  the 
tangent  at  P^.  Hence,  to  get  the  equation  of  the  normal 
at  any  given  point,  it  is  only  necessary  to  write  the  equation 
of  the  tangent  at  this  point  (Art.  84),  and  then  the  equa- 
tion of  a  line  perpendicular  to  this  tangent  (Arts.  53,  62) 
and  passing  through  the  given  point.  Thus  the  equation 
of  the  normal  to  the  circle 

x^  +  f  +  2ax+2Fi/  +  O=0,     .     .     .     (1) 
at  the  point  P^  =(ix-^,  «/i),  is 

The  coordinates  —  Gr  and  —  P  of  the  center  of  the  given 
circle  (1)  satisfy  equation  (2) ;  hence^  every  normal  to  a  circle 
jmsses  through  the  center  of  the  circle. 

If  the  center  of  the  circle  be  at  the  origin,  then  (7  =  0, 
P  =  0,  and  Q=—r'^,  and  the  equation  (2)  of  the  normal 
becomes 

which  reduces  to  x^y  —  xy^  =  0,  —  an  equation  which  could 
have  been  derived  for  the  circle  x'^  +  y'^  =  r^  in  precisely  the 
same  way  that  equation  (2)  was  derived  from  equation  (1). 

EXERCISES 

1.  Derive,  by  the  secant  method,  the  equation  of  the  tangent  to  the 
circle  x^  -}-  y^  =  2  rx,  the  point  of  contact  being  P^  =  (x^,  y^ . 

2.  Write  the  equation  of  the  tangent  to  the  circle  : 
(a)    x^  +  ?/2  =  25,  the  point  of  contact  being  (3,  4)  ; 

(^)  a;2  +  ?/2  -  3  a;  +  10  ?/  =  15,  the  point  of  contact  being  (4,  -11)  ; 
(y)   (x  -  2)2  +(y  -  3)2  =  10,  the  point  of  contact  being  (5,  4) ; 
(8)   3  a:2  +  3  2/2  -  2  2/  -  4  a:  =  0,  the  point  of  contact  being  (0,  0). 


85-86.] 


THE  CIRCLE 


149 


3.  Find  the  equation  of  the  normal  to  each  of  the  cu'cles  of  Ex.  2, 
through  the  given  point. 

4.  A  tangent  is  perpendicular  to  the  radius  drawn  to  its  point  of 
contact.  By  means  of  this  fact,  derive  the  equation  of  the  tangent  to 
the  circle  {x  —  ay^{y  —  by"  =  r'^  at  the  point  {x^,y^)  (cf.  equation  [36]). 

5.  From  the  fact  that  a  normal  to  a  circle  passes  through  its  center, 
find  the  equation  of  the  normal  to  the  circle  x^  +  y^  —  6 a;  +  8^  +  21  =0 
at  the  point  (1,  "4). 

6.  Find  the  equations  of  the  two  tangents,  drawn  through  the  ex- 
ternal point  (11,  3)  to  the  circle  a;^  ■\-  if-  —  40. 

Suggestion.     Use  the  equation  of  the  tangent  in  terms  of  its  slope. 

7.  What  is  the  equation  of  the  circle  whose  center  is  at  the  point 
(5,  3),  and  which  touches  the  line  3a;  +  2?/  —  10  =  0? 

X       v 

8.  Under  what  condition  will  the  line  -  +  ;-  =  !   touch   the   circle 

a      0 

a;2  +  ?/2  =  7'2  9 

9.  Find  the  equation  of  a  circle  inscribed  in  the  triangle  whose  sides 

are  the  lines  x  =  0,  y  =  0,  and  _  _(-  ^  =  1 . 

a     b 

10.  Solve  Ex.  6  by  assuming  x-^^  and  y-^  as  the  coordinates  of  the  point 
of  contact,  and  then  finding  their  numerical  values  from  the  two  equa- 
tions which  they  satisfy. 


86.  Lengths  of  tangents  and  normals.  Subtangents  and 
subnormals.  The  tangent  and  normal  lines  of  any  curve 
extend  indefinitely"  in  both 
directions ;  it  is,  however, 
convenient  to  consider  as  the 
length  of  the  tangent  the 
length  TP^,  measured  from 
the  point  of  intersection  (T) 
of  the  tangent  with  the  x- 
axis  to  the  point  of  tangency 
(Pj),  and  similarly  to  consider  as  the  length  of  the  normal 
the  length  P^N^  measured  from  P^  to  the  point  of  intersec- 
tion (iV)  of  the  normal  with  the  2;-axis. 


150 


AJSTALYTIC  GEOMETRY 


[Ch.  VII. 


The  subtangent  is  tlie  length  TM,  where  M  is  the  foot  of 
the  ordinate  of  the  point  of  tangency  P^ ;  and  the  subnormal 
is  the  corresponding  length  MN.  As  thus  taken,  the  sub- 
tangent  and  the  subnormal  are  of  the  same  sign  ;  ordinarily, 
however,  one  is  concerned  merely  with  their  absolute  values, 
irrespective  of  the  algebraic  sign.  The  subtangent  is  the 
projection  of  the  tangent  length  on  the  2:-axis,  and  the  sub- 
normal is  the  like  projection  of  the  normal  length. 


87.  Tangent  and  normal  lengths,  subtangent  and  subnor- 
mal, for  the  circle.  The  definitions  given  in  the  preceding 
article  furnish  a  direct  method  for  finding  the  tangent  and 
normal  lengths,  as  well  as  the  subtangent  and  subnormal, 
for   a    circle.      ^•^.,    to    find    these    values    for   the    circle 

x^-\- 1/^  =  25,  and  correspond- 
ing to  the  point  of  contact 
(3,  4),  proceed  thus: 

The  equation  of  the  t^ni- 
gentP^Tis  (Art.  84) 


Fig.  68. 


hence  the  a:-intercept  of  this 
tangent,    ^.e.,    OT,    =  ^^- ; 

therefore  the  subtangent   TM,   which  equals   OM  —  OT^  is 

3  —  ^,  ^.e.,  —  5J.     The  tangent  length 


TP^  =  ^MT^  +  MPl''  =  V(J^)2  +  42  =  6|. 

To  find  the  normal  length,  and  the  subnormal,  first  write 
the  equation  of  the  normal  at  the  point  (3,  4);  it  is  (Art. 
86)  4:x  —  Sy  =  0.  Hence  its  a;-intercept  is  zero,  and  the 
subnormal,  MO  in  this  case,  is  —  3  ;  the  normal  length  P^  0 
is  5. 


86-88] 


THE   CIRCLE 


151 


Similarly,  corresponding  to  the  point  (x^,  y-^  on  the  circle 


a;2  _|.  ^2  __  ^2^    ^]^Q    subtangent    =  —  ^,    the   tangent  length 


X. 


^^] 


=  -^,   the  subnormal   =  —  x^^   and  the  normal  length   =  r. 

The  derivation  of  these  values  is  left  as  an  exercise  for  the 
student,  as  is  also  the  derivation  of  the  corresponding 
expressions  for  the  circle  x^  -\-  y^  -\-1  Gx  +  2  Fy  +  C^  =  0,  the 
point  of  contact  being  {x^^  y^. 

EXERCISES 

Find  the  lengths  of  the  tangent,  subtangent,  normal,  and  subnormal, 

1.  for  the  point  (4,  -11)  on  the  circle  a:^  +  ?/2  —  3  a;  +  10  ?/  =  15 ; 

2.  for  the  point  (1,  3)  on  the  circle  x'^  +  y'^  —  V) x  =0] 

3.  for  the  point  whose  abscissa  is  V7  on  the  circle  a;-  +  y^  =  25. 

4.  The  subtangent  for  a  certain  point  on  a  circle,  whose  center  is  at 
the  origin,  is  5i,  and  its  subnormal  is  3.  Find  the  equation  of  the  circle, 
and  the  point  of  tangency, 

88.  To  find  the  length  of  a  tangent  from  a  given  external 
point  to  a  given  circle.  Let  P-^  =  {x-^,  y^  be  the  given 
external  point,  and  let 

a;2  +  y2  _|_  2  fe  +  2  ^^  +  C  =  0 

be  the  given  circle.  The  center  of  this  circle  (Art.  79)  is 
(~(r,  ~-F),  and  its    radius   is 

center  K^   draw  the  tangent 
Pj§,  and  also  the  radius  KQ. 

Then  P^  =  KP^  -  KQ^ ; 

but 

(Art.  26) 
and 


o 


Fig.  69. 


^=  (72-hP2_^. 


(Art.  79) 


152  ANALYTIC   GEOMETBT  [Ch.  VII. 

i.e.,  the  square  of  the  length  of  the  tangent  from  a  given 
external  point  to  the  circle  x^  +  7/^4-2  Gix  +  2  Fy  +  C  =  0  * 
is  obtained  by  writing  the  first  member  only  of  this  equation.^ 
and  substituting  in  it  the  coordinates  of  the  given  point. f 

89.   From  any  point  outside  of  a  circle  two  tangents  to  the 
circle  can  be  drawn,      (a)  Let  the  equation  of  the  circle  be 

0:2  +  ^2^^2,  .  .  .  (1) 

then  (Art.  83)  the  line 

y  =  mx  +  r Vl  +  ni^    .  .  .  (2) 

is,  for  all  values  of  m,  tangent  to  this  circle.  Let  P^={x^,  ?/j) 
be  any  given  point  outside  the  circle  (1);  then  the  tangent 
(2)  will  pass  through  P^  if,  and  only  if,  m  be  given  a  value 
such  that  the  equation 


y^  =  mxi  +  rVl  -f  711^     .  .  .       (3) 

shall  be  satisfied. 

Transposing,  squaring,   and  rearranging  equation  (3),  it 

is  clear  that  it  will  be  satisfied  if,  and  only  if,  m  is  given  a 

value  such  that  the  equation 

(r^  —  x-^^m^  +  2  x-^y^m  +  r^  —  y^  —  0 

is  satisfied;  i.e.,  equation  (3)  is  satisfied  if,  and  only  if. 


^  ^  -  ^^y^  ±  ^v^i    +  Va   -  ^'\      ...      (4) 


—  x^y-^  ±  r^x^  +  yf  —  r^ 
r^  —  x^ 

Equation  (4)  gives  two,  and  only  two,  real  values  for  m 
when  {xy^  y^  is  outside  of  the  circle,  for  then  x^  +  y^  —  r^  is 

*  If  the  circle  is  given  by  the  equation  Ao^  +  Aip-  +  2  G^x  -f  2  F?/  +  C  =  0, 
it  must  first  be  divided  by  A  before  applying  this  theorem. 

t  The  expression  x^  +  ?/i^  +  2  Gx\  +  2  Fy\  +  (7  is  called  the  power  of  the 
point  Pi=  (xi,  yi)  with  regard  to  the  circle  x'^  +  y"^  -\-  2.  Gx  -\-  '2.  Fy  -{-  (7  =  0. 


88-89.]  THE  CIRCLE  153 

positive  (Art.  78,  foot-note)  ;  these  values  of  m,  being  sub- 
stituted in  turn  in  equation  (2),  give  the  two  tangents 
through  Pj  to  the  circle  (1). 

If  Pj  is  071  the  circle  (1),  then  x-^  -{-  7/^  —  r^  =  0;  hence  the 
two  values  of  m  from  equation  (4)  coincide,  and  the  two 
tangents  also  coincide,  i.e.,  there  is  in  this  case  but  07ie 
tangent.  If  P^  is  within  the  circle,  then  the  two  values 
of  m  from  equation  (4)  are  both  imaginary  and  no  tangent 
through  Pj  can  be  drawn  to  the  circle  (1).* 

If  either  value  of  m  from  equation  (4)  is  substituted  in 
equation  (2),  and  then  equations  (2)  and  (1)  are  considered 
as  simultaneous  and  solved  for  x  and  ^,  the  coordinates  of 
the  corresponding  point  of  contact  are  obtained. 

Note.  The  properties  of  the  equations  of  the  Une  and  circle  have  thus 
established  a  geometric  property  of  the  circle  [cf.  Art.  31,  (III)]. 

(^)  If  the  equation  of  the  given  circle  had  been 

x^  +  f-\-2ax  +  2F^+O  =  0,     .     .     .     (5) 

it  could,  by  Art.  71,  have  been  transformed  to  new  axes 
through  its  center  (~(7,  ~P)  and  parallel  respectively  to 
the  given  axes ;  its  equation  would  thus  have  become 


X 


12    I    ^,/2  _  J2, 


+  y2  =  r2,  .  .  ,  (6) 

where  x^  and  y^  refer  to  the  new  axes. 

This  transformation,  however,  leaves  the  circle  and  all  its 
intrinsic  properties  unchanged  ;  but  (a)  applies  to  circle  (6), 
hence  it  is  proved  that  circle  (5),  which  is  circle  (6)  merely 
referred  to  other  axes,  has  the  same  properties. 

*  These  conclusions  may  also  he  stated  thus :  if  Pi  is  outside  of  the 
circle,  equation  (4)  gives  two  real  and  distinct  values  for  m  ;  corresponding 
to  these  there  are  two  real  and  distinct  tangents  ;  if  Pi  is  on  the  circle,  the 
two  values  of  m  are  real  but  coincident,  and  there  are  two  real  but  coincident 
tangents  ;  if  Pi  is  inside  of  the  circle,  the  two  values  of  m  are  imaginary, 
and  the  two  corresponding  tangents  are  therefore  also  imaginary. 


154 


ANALYTIC   GEOMETRY 


[Ch.  VIL 


90.  Chord  of  contact.  If  two  tangents  are  drawn  from  any 
external  point  to  a  circle,  the  line  joining  the  two  corre- 
sponding points  of  tangency  is  called  the  chord  of  contact  for 
the  point  from  which  the  tangents  are  drawn. 

The  equation  of  this  chord  of  contact  may  be  found  by 

first  finding  the  points  of  tan- 
gency and  then  writing  the 
equation  of  the  straight  line 
through  those  two  points.  It 
may,  however,  be  found  more 
briefl}^,  and  much  more  ele- 
gantly, as  follows : 

Let  P^  =  (tj,  y^)  be  the 
given  external  point  from 
which  the  two  tangents  are 
drawn ;  and  let  T^  =  (x^^  y^  and  T^  =  (.rg,  ^3)  be  the  points 
of  tangency  on  the  circle 

x^  +  f  +  2ax-\-2Fy-\-C=0;  .  .  .  (1) 
it  is  required  to  find  the  equation  of  the  line  passing  through 
T2  and  2^3.     The  equation  of  the  tangent  at  T^  is  (Art.  84) 

^2^  +  I/2I/  +  ^(^  +  ^2)  +  ^(1/  +1/2)  +  C^  =  0,  .  .  .  (2) 
and  the  equation  of  the  tangent  at  T^  is 

But  each  of  these  tangents  passes  through  the  point  Pj  ; 
hence  its  coordinates,  x^  and  ^j,  satisfy  equations  (2)  and  (3), 
therefore 

^1^2  +  ^1^2  +  ^(^1  +  ^2)  +  ^(1/1  +  ?/2)  +  ^  =  0,  .  .    .    (4) 

and      x^x^  +  y^y^  -f  G-Qx^  +  x^}  -f  F(y^  +  2/^^-\- C  =0.  .  .  .  (5) 
Equations  (4)  and  (5),  however,  assert  respectively  that 
(^2'  ^2)  ^^^  C^3^  ^s)  ^^^  points  on  the  locus  of  the  equation 
^i^  +  ^i^  +  ^(^i  +  ^)  +  ^(^i  +  «/)+C  =  0.  .   .   .   (6) 


90.]  THE  CIRCLE  155 

But  equation  (6)  is  of  the  first  degree  in  the  two  varia- 
bles X  and  ?/,  hence  (Art.  57)  its  locus  is  a  straight  line,  and, 
since  it  passes  through  both  T-^^  =  (x2'>  I/2)  ^^^  ^3=(^3'  ^3)' 
it  is  the  equation  of  the  chord  of  contact ; 

i.e.,  x^x  +  y^y  +  G-{x  +  x^)  -^F{y  +  y^)  +  C  =  0  .  .  .  [37] 
is  the  equation  of  the  chord  of  contact  corresponding  to  the 
external  point  P^^(x^^  y^. 

It  is  to  be  noticed  that  if  Pj  is  on  the  circle,  then  the  two 
tangents  drawn  through  it  coincide  with  each  other  and  with 
the  chord  of  contact ;  the  equation  of  the  chord  of  con- 
tact [37]  then  becomes  the  equation  of  the  tangent  at  P^,  as 
it  should  (cf.  equation  [36]). 

If,  then,  (a^j,  yj)  is  a  point  on  the  circle  (1),  equation  [37] 
is  the  equation  of  the  tangent  to  the  circle  at  that  point ;  if, 
on  the  other  hand,  (x^,  y^  is  outside  of  this  circle,  then 
equation  [37]  is  not  the  equation  of  a  tangent,  but  of  the 
chord  of  contact  corresponding  to  that  external  point. 

EXERCISES 

1.  Find  the  length  of  the  tangent  from  the  point  (8,  10)  to  the  circles  : 

2.  (a)  Write  the  equation  of  the  chord  of  contact  corresponding  to 
the  point  (5,  6)  for  the  circle  x'^  +  y'^  —  Qx  —  4.y  =  8. 

(/3)  Find  the  coordinates  of  the  points  in  which  this  chord  cuts  the 
circle. 

(y)  Write  the  equations  of  the  tangents  to  the  circle  at  these  points 
of  intersection  ;  show  that  these  lines  pass  through  the  given  point  (5,  6). 

3.  By  the  method  of  exercise  2,  find  the  equations  of  the  tangents 
drawn  to  the  circle  (o  x  —  2)^  +  (3  ?/  +  5)2  —  4^  from  the  origin ;  from  the 
point  (1,  2). 

4.  Find  the  locus  of  a  point  from  which  the  tangents  drawn  to  the 
two  circles 

2  ^2  +  2  ?/2  -  10  X  +  14  y  +  35  =  0     and     x2  +  ?/2  =  9 

are  of  equal  length.  Show  that  this  locus  is  a  straight  line  perpendicular 
to  the  line  joining  the  centers  of  the  given  circles. 


156 


ANALYTIC  GEOMETRY 


[Ch.  VII. 


5.  For  what  point  is  the  line  3  a;  +  4?/  =  7  the  chord  of  contact  with 
regard  to  the  circle  x-  +  ^^  =  14  ? 

6.  Find  the  chord  of  contact  for  the  circle  x^  +  y-  =  25,  corresponding 
to  the  point  (3,  7) ;  to  the  point  (3,  2). 

7.  By  means  of  the  equation  y  —  y-^  =  7n(x—  x-^)  prove  that  two  tan- 
gents can  be  drawn  through  the  external  point  (.r^^,  y^)  to  the  circle 
whose  equation  is  x'^  -\-  y^  =  r^- 

8.  Solve  (/3)  and  (y),  of  exercise  2,  by  means  of  the  equation 

y  —  Q  =  m(x  —  5). 

91.  Poles  and  Polars.  If  through  any  given  pomt 
Pj  =  (2:p^j),  outside,  inside,  or  on  the  circle,  a  secant  is 

drawn,  meeting  the  circle  in  two 
points,  as  Q  and  R^  and  if  tan- 
gents are  drawn  at  Q  and  R^  they 
will  intersect  in  some  point  as 

The  locus  of  P',  as  the  secant 
revolves  about  Pj,  is  called  the 
polar  of  Pj  with  regard  to  the 
circle ;  and  Pj  is  the  pole  of  that 
locns.  It  will  be  proved  in  the 
next  article  that  the  locus  of  P' 
is  a  straiglit  line  whose  equation  is  of  the  same  form  as  that 
of  the  tangent  (Art.  84),  and  as  that  of  the  chord  of  contact 
(Art.  90)  already  found. 

92.  Equation  of  the  polar.  Let  P^  =  (x^,  y{)  be  the  given 
point,  the  equation  of  whose  polar,  with  regard  to  the  circle 

x^  +  y'^  +  2ax^2Fy-{-C=0,  .  .  .  (1) 
is  sought.  Also  let  P^QR  be  any  position  of  the  secant 
through  Pj,  and  let  the  tangents  at  Q  and  R  intersect  in 
P'=(a;',  y);   then  the  equation  of  P^QR  (Art.  90)  is 

x'x  +  y'y  +  a(x^x'^-{-F(iy  +  y')+C=^.    ...   (2) 


Fig.  71. 


90-93.] 


THE   CIRCLE 


157 


Since  P^  is  on  this  line,  therefore 

^i^'  +  yi/  +  ^(^i  +  ^0  +  -^(yi  +  yO  +  C^=0.  .  .  .  (3) 

Equation  (3)  asserts  that  the  coordinates,  x'  and  y\  of 
P'  satisfy  the  equation 

.    ^i^  +  yi^  +  ^(^  +  ^i)  +  -^(y  +  yi)+C^=0;    .   .   .    [38] 

i.e.^  this  variable  point  P'  always  lies  on  the  locus  of  equa- 
tion [38] ;  in  other  words,  [38]  is  the  equation  of  the  polar 
of  P-^  with  regard  to  the  circle  (1). 

Moreover,  since  equation  [38]  is  of  the  first  degree  in  the 
variables  x  and  y,  therefore  (Art.  57)  its  locus  is  a  straight 
line ;  that  is,  the  polar  of  any  given  pointy  with  regard  to  any 
given  circle^  is  a  straight  line. 

That  equations  [36]  and  [37]  have  the  same  form  as  equa- 
tion [38]  is  due  to  the  fact  that  the  tangent  and  the  chord 
of  contact  are  only  special  cases  of  the  polar. 

93.  Fundamental  theorem.  An  important  theorem  con- 
cerning poles  and  jDolars  is  :  If  the  polar  of  the  point  P^, 
with  regard  to  a  given  circle^ 
passes  through  the  point  P^., 
then  the  polar  of  P^  passes 
through  P^.  Let  the  equa- 
tion of  the  given  circle  be 

x^  +  y'-  +  2ax  +  2Fy 

-h(7=0,  .     .     .     (1) 

and  let  the  two  given  points 

be  Pi  =  Cx^,  ^i), 

and        P2  =  (.^2^1/2)1 

then  (Art.  92)  the  equation  of  the  polar  of  P^  is 

x^x+y^y  +  a(x-\-x^)  +  F(y-^y^)+C=0.   ...   (2) 


158  ANALYTIC   GEOMETRY  [Ch.  VIL 

If  this  line  passes  through  P^,  then 

But  the  equation  of  the  j)olar  of  P^  (Art.  92)  is 

x^x  +  y,j  +  a(x  +  x^^^-Fiy  +  y,J  +  O=0,   ...   (4) 

and  equation  (3)  proves  that  the  locus  of  equation  (4)  passes 
through  Pj,  which  establishes  the  theorem. 

EXERCISES 

1.  Find  the  polar  of  the  point  (6,  8)  with  reference  to  the  circle 
x^  +  y^  =  14. 

2.  Find  the  x:>olar  of  the  pohit  (1,  2)  with  regard  to  the  circle 
x^  +  y^  -\-  ^  X  —  Q  y  —  10. 

3.  Find  the  pole  of  the  line  4x  +  6?/  =  7,  and  of  the  line  aa:  +  %  — 1  =  0, 
with  regard  to  the  circle  x'^  +  ?/^  =  35. 

4.  Find  the  equations  of  the  two  tangents  to  the  circle  x^  +  y^  =  65 
from  the  point  (4,  7);  from  the  point  (11,  3). 

5.  Show  that  if  the  polar  of  (Ji,  k)  with  respect  to  the  circle  x'^+y'^  =  c^ 
touch  the  circle  4  (x'^  +  y^)  =  c^,  then  the  pole  (h,  k)  will  lie  on  the  circle 

■x^  +  y2  —  4(^2. 

6.  Show  that  the  pole  of  the  line  joining  (5,  7)  and  (~11,  1)  is  the 
point  of  intersection  of  the  polars  of  those  two  points  with  reference  to 
the  circle  x'^  +  ?/-  =  100. 

7.  Find  the  pole  of  the  line  2x  —  3y  =  0  with  respect  to  the  circle 
x^  +  y^-  =  9. 

8.  Show  what  specialization  of  a  polar  converts  it  into  a  chord  of 
contact,  and  what  further  specialization  converts  it  into  a  tangent. 

94.  Geometrical  construction  for  the  polar  of  a  given  point, 
and  for  the  pole  of  a  given  line,  with  regard  to  a  given  circle. 

Since  tlie  relation  between  a  polar  and  its  pole  (see  def. 
Art.  91)  is  independent  of  the  coordinate  axes,  therefore 
the  given  circle  may,  without  loss  of  generality,  be  assumed 
to  have  its  center  at  the  origin. 

If  P^  =  (a?j,  ^j)  is  any  given  point,  and 

a^-i-f  =  r^  .         .  .  (1) 


93-94.] 


THE  CIRCLE 


159 


is  a  given  circle,  whose  center  is  at  the  point  0,  then  the 
equation  of  OP^  (Art.  51)  is 

y^x  —  x^y^O.       ...  (2) 


/^ 

Y 

^ 

/ 

Xk 

/ 

Py 

\ 

\ 

I    0 

X 

\ 

\ 

\  ^ 

^^ 

) 

\ 

^ — 

Fig. 

73.^ 

Fig.  73; 


Let  LL^  be  the  polar  of  P^,  with  regard  to  the  given 

circle,  and  let  it  meet   OP^  in   K.     The   equation  of  LL-^ 

(Art.  92)  is 

x^x-\-y^y=-r^,        .  .  .  (3) 

Equations  (2)  and  (3)  show  (Art.  62)  that  LL^  and  OP^ 
are  perpendicular  to  each  other;  ^.e.,  the  line  joinmg  the 
given  point  P^  to  the  center  of  the  circle  is  perpendicular  to 
the  polar  of  P^  with  regard  to  the  circle. 

The  distance  (OIC)  from  the  origin  to  the  line  LL^ 
(Art.  04)  is 


■\/x^  4-  y-^ 
and  the  length  of  OP-^  (Art.  26)  is 


Vx^+y} 


C^) 


(5) 


therefore 


OK'  OP^^ 


,,.2 


■\/x^  +  y-^  =  /■■ 


-Vx^^  +  yi^ 

Hence,  to  construct,  with  regard  to  a  given  circle,  the 
polar  of  any  given  point  P^,  join  that  point  to  the  center  of 
the  circle,  then  on  OP^  (produced  if  necessary)  find  a  point 
K  such  that  the  rectangle  OP^  •  OK  is  equal  to  the  square 


160  ANALYTIC  GEOMETRY  [Ch.  VII. 

on  the  radius  of  the  circle,  and  through  K  draw  a  line 
perpendicular  to  OPj ;  this  line  is  the  required  polar. 

Similarly  the  pole  may  be  constructed,  if  the  polar  and 
the  circle  are  given. 

95.  Circles  through  the  intersections  of  two  given  circles. 
Given  two  circles  whose  equations  are 

x'  ^  f  -^  2a,x  -{-  2F,i/  ^  C,  =  0,    .     .     .     (1) 

and  x'  +  f-{-'2  a^x  +  2F^y -\-  0^  =  0.    ,     .     .     (2) 

These  circles  intersect,  in  general,  in  two  finite  j)oints 
P-^  =  (xi,  7/i)  and  P2  =  Qx2,  ?/2),  and  (Art.  41)  the  equation 

x'^f^  +  2a,x+2F,y-\-  C\ 

+  k(x'  +  f^  +  2a2X^2F.^-i-CO  =  0,    ...   (3) 

where  k  is  any  constant,  represents  a  curve  which  passes 
through  these  same  points  Pi  and  P2. 

The  locus  of  equation  (3)  is,  moreover,  a  circle  (Art.  79) ; 
hence,  a  series  of  different  values  being  assigned  to  the  param- 
eter ^,  equation  (3)  represents  what  is  called  a  "family" 
of  circles ;  each  one  of  these  circles  passing  through  the  two 
points  Pi  and  P2  in  which  the  given  circles  (1)  and  (2) 
intersect  each  other. 

96.  Common  chord  of  two  circles.  If  in  equation  (3), 
Art.  95,  the  parameter  Jc  be  given  the  particular  value 
—  1,  the  equation  reduces  to 

2  ((7i  -  a2')x  +  2(Fi-  F2^y  +  (7i  -  a=0,  .  .  . .  (4) 

which  is  of  the  first  degree,  and  therefore  represents  a 
straight  line  ;  but  this  locus  belongs  to  the  family  repre- 
sented by  equation  (3)  of  Art.  95,  hence  it  passes  through  the 
two  points  Pi  and  P2  in  which  the  circles  (1)  and  (2)  inter- 
sect. This  line  (4)  is,  therefore,  the  common  chord*  of 
these  circles. 


94-97.]  THE   CIRCLE  161 

To  obtain  the  equation  of  the  common  chord  of  two  given  circles  it  is, 
then,  only  necessary  to  eliminate  the  terms  in  x^  and  y^  between  their 
equations.     E.g.,  to  find  the  common  chord  of  the  circles 

2a:2  +  2y2+    3a:+    5?/-    9  =  0,  .       .       .       (a) 

and  6x2 +  6  3/2 +  11  a: +  13  2/ -23  =  0,  •       •       •      (/?) 

multiply  equation  (a)  by  3  and  subtract  the  result  from  equation  {^) ; 
this  gives 

x-?/  +  2  =  0,  .  .  .  (y) 

as  the  equation  of  the  common  chord  of  the  given  circles. 

This  result  may  be  verified  by  finding  the  points  of  intersection 
(Art.  89)  of  the  circles  (a)  and  (/?),  and  then  writing  the  equation  of 
the  straight  line  through  those  two  points. 

Since  the  common  chord  of  two  circles  intersects  each  of  these  circles 
in  the  points  in  which  they  intersect  each  other,  therefore  the  points 
of  intersection  of  two  circles  may  be  found  by  finding  the  points  in 
which  their  common  chord  intersects  either  of  them.  E.g.,  to  find  the 
points  in  which  the  circles  (a)  and  (y8)  intersect  each  other,  it  is  only 
necessary  to  find  the  points  in  which  (y)  cuts  either  (a)  or  (/3). 

97.  Radical  axis  ;  radical  center.  The  line  whose  equation 
is  obtained  by  eliminating  the  x^  and  y"^  terms  between  the 
equations  of  two  given  circles,  as  in  Art.  96,  whether  the 
circles  intersect  in  real  points  or  not,  is  called  the  radical  axis 
of  the  two  circles.  If  the  two  given  circles  intersect  each 
other  in  real  points,  then  this  line  is  also  called  their  com- 
mon chord  ;  that  is,  the  common  chord  of  two  circles  is  a 
special  case  of  the  radical  axis  of  two  circles. 

*  Equation  (3)  of  Art,  95,  which  for  every  value  of  k  represents  a  circle 
passing  through  the  two  points  in  which  the  given  circles  (I)  and  (2)  inter- 
sect, may  be  written  in  the  form 

x2  +  y2  +  2^L^tM.x  +  2^1+l^^+^^hl^=0. 
1  +  ^-  i  +  ic     ^^     I  +k 

The  coordinates  of  the  center  of  this  circle  are  (Art.  79) 

\+k  1 +k 

If  then  k  be  made  to  approach  —1,  both  of  these  coordinates  approach 
infinity,  but  the  circle  always  passes  through  the  two  fixed  points  in  which 
the  given  circles  intersect ;  hence  the  common  chord  of  two  given  circles 
may  be  regarded  as  an  infinitely  large  circle  whose  center  is  at  mfinity. 

TAN.   AN.    GEOM. 11 


162 


ANALYTIC  GEOMETRY 


[Ch.  VII. 


Three  circles,  taken  two  and  two,  have  three  radical  axes. 
It  is  easily  shown  that  these  three  radical  axes  pass  through 
a  common  point ;  this  point  is  called  the  radical  center  of  the 
three  circles. 

EXERCISES 

1.  Find  the  equation  of  the  common  chord  of  the  cu'cles 

2.  Find  the  point  of  intersection  of  the  circles  in  exercise  1,  and  the 
length  of  their  common  chord. 

3.  Find  the  radical  axis,  and  also  the  length  of  the  common  chord, 
for  the  circles  x^  +  y^  +  ax  -i-  by  -\-  c  =  0,  x^  -i-  y'^  -\-  bx  -\-  ay  +  c  =  0. 

4.  Find  the  radical  center  of  the  three  circles 

a:^  +  ?/2  +  4  a:  +  7  =  0, 

2  (a;2  +  2/2)  +  3 2-  +  5y  +  9  =  0, 

x^  +  y-  +  y  —  0. 

5.  Show  that  tangents  from  the  radical  center,  in  exercise  4,  to  the 
three  circles,  respectively,  are  equal  in  length. 

6.  Prove  analytical!}^  that  the  tangents  to  two  circles  from  any  point 
on  their  radical  axis  are  equal. 

7.  Find  the  polar  of  the  radical  center  of  the  circles  in  exercise  4, 
with  respect  to  each  circle. 

8.  Prove  analytically  that  the  three  radical  axes  of  three  circles,  the 
circles  being  taken  in  pairs,  meet  in  a  common  point. 

98.   The  equation  of  a  circle :  polar  coordinates.     Let  OB 

be  the  initial  line,  0  the  pole,  C=(p^,  6{)  the  center  of  the 

circle,  r  its  radius,  and  P  =  (p,  ^) 
any  point  on  the  circle.  Draw  0(7, 
OP,  and  OP  ;  then,  by  trigonometry, 

r^  =:  p^  +  p^^  —  2  pp^  cos  (^  —  ^j),  i.e., 

/32_2/)i/9COS(6>-  (9i) 

+  ^^2  _  ^2  ^  0,  .   .   .    [39] 

which  is  the  equation  of  the  given 
circle. 


Fig.  74. 


97-99.]  THE  CIRCLE  163 

Depending  upon  the  relative  positions  of  the  polar  axis, 
the  pole,  and  the  center  of  the  circle,  equation  [39]  has 
several  special  forms  : 

(a)  If  the  center  is  on  the  polar  axis,  then  6-^  =  0,  and 
equation  [39]  becomes 

p^  —  2  p^p  cos  ^  +  p^2  _  ^2  _  0  ; 

(/3)  If  the  j)ole  is  on  the  circle,  then  p-^  =  r,  and  equa- 
tion [39]  becomes 

/3- 2rcos(6'-6'i)=  0; 

(7)  If  the  pole  is  on  the  circle  and  the  polar  axis  a  diame- 
ter, then  pi  =  r  and  0^  =  0,  and  equation  [39]  becomes 

p  —2r  cos6  =  0  ; 

(5)  If  the  center  is  at  the  pole,  then  pi  =  0  and  equation 

[391  becomes 

L     J  p  =  r. 

99.   Equation  of  a  circle  referred  to  oblique  axes.     Let  the 

axes  OX  and  Oy  be  inclined  at  an  angle  co  ;  let  C  =  {h,  k^ 
be   the    center   of   the    circle,    r  _ 

its  radius,  and  P  =  (ic,  ^)  any 
point  on  the  circle.  Draw  the 
ordinates  M^Q  and  MP^  connect 
C'and  P,  and  draw  CHL  paral- 
lel to  the  a;-axis  ;   then 

Fig.  75. 

-\- 2  OR '  RP  COS  CO  ; 

hence  r^  =  (x  —  h)"^ -\- Qy  —  k^  +  2(x  —  K){y  —  k)  cos  co, 
{.e.,(x-hy+(y-ky^-\-2<^x-h)(2j-k)Gosco-r^  =  0;,  ..[40] 
which  is  the  equation  of  the  given  circle. 


164  ANALYTIC  GEOMETRY  [Ch.  VII. 

It  is  to  be  observed  that  this  equation  [40]  is  not  of  the 
form 

x^^yij^2ax  +  2Fy  +  (7=  0, 

which  was  discussed  in  Art.  79  ;  it  differs  from  that  equa- 
tion in  that  it  contains  an  a^y-term.  If,  however,  the  axes 
are  rectangular,  as  in  Art.  79,  then  cos  o)  =  0,  and  equation 
[40]  reduces  to  the  standard  form  of  Art.  79,  viz. : 

a;2  ^  ^2  _|_  2  (7a;  +  2 IV  +  (7=  0, 

which  is  a  special  case  of  equation  [40]. 

100.   The  angle  formed  by  two  intersecting  curves.    By  the 

angle  between  two  intersecting  curves  is  meant  the  angle 
formed  by  the  two  tangents,  one  to  each  curve,  drawn 
through  the  point  of  intersection. 

Hence  to  find  the  angle  at  which  two  curves  intersect,  it 
is  only  necessary  to  find  the  point  of  intersection,  then  to 
find  the  equations  of  the  tangents  at  this  point,  one  to  each 
curve,  and  finally  to  find  the  angle  formed  by  these  tangents. 


EXERCISES 

1.  Find  the  polar  equation  of  the  cii'cle  whose  center  is  at  the  point 

(  7,  -T  j  and  whose  radius  is  10 ;  determine  also  the  points  of  its  inter- 
section with  the  initial  line. 

2.  Find  the  polar  equation  of  a  circle  whose  center  is  at  the  point 

(  15,  -  J  and  whose  radius  is  10.     Find  also  the  equations  of  the  tangents 
to  the  circle  from  the  pole. 

3.  A  circle  of  radius  3  is  tangent  to  the  two  radii  vectores  which 
make  the  angles  60°  and  120°  with  the  initial  line :  find  its  polar  equa- 
tion, and  the  distance  of  the  center  from  the  origin. 

4.  Find  the  equation  of  a  circle  of  radius  5,  with  center  at  the  point 
(2,  3),  if  o)  is  60°. 

5.  Find  the  equation  of  a  circle  of  radius  2,  with  center  at  the  origin, 
if  (o  is  120°. 


99-100.]  THE   CIRCLE  '  1(J5 

6.  Determine  the  equation  of  tlie  circle  circumscribing  an  equilateral 
triangle,  —  the  coordinate  axes  being  two  sides  of  the  triangle. 

7.  A  circle  is  inscribed  in  a  square.  What  is  its  equation,  if  a  side 
and  adjacent  diagonal  of  the  square  are  chosen  as  the  y-  and  ^--axis, 
respectively?     What  are  the  coordinates  of  the  points  of  tangency? 

8.  Find  the  angle  at  which  the  circle  x^  +  3/2  =  9  intersects  the  circle 
(x-  —  4)-  +  y-  —  2  1/  =  15.  At  what  angle  does  the  second  of  these  circles 
meet  the  line  x  +  2  y  =  ^1 

EXAMPLES    ON    CHAPTER   VII 

1.  Find  the  equation  of  the  cipcle  circumscribing  the  triangle  whose 
vertices  are  at  the  points  (7,  23^"1)  ~4),  and  (3,  3).  What  is  its  center? 
its  radius  ? 

2.  Determine  the  center  of  the  circle 

(^x  +  a)2  +  (?/  +  &)2  =  a2  +  i\ 

What  family  of  circles  is  represented  by  this  equation,  if  a  and  h 
vary  under  the  one  restriction  that  a^  +  &2  is  to  remain  constant  ? 

3.  What  must  be  the  relations  among  the  coefficients  in  order  that 

the  circles 

x2  +  2/2  +  2Gia;  +  2i^,?/+  C^  =  0, 

and  ^2  +  ?/2  +  2  GgX  +  2  F^y  +  Cg  =  0, 

shall  be  concentric  ?  that  they  shall  have  equal  areas  ? 

4.  Under  what  limitations  upon  the  coefficients  is  the  circle 

Ax"^  +  Ay'^  +  Dx^  Ey  -V  F=0 
tangent  to  each  of  the  axes  ? 

5.  Find  the  equation  of  the  circle  which  has  its  center  on  the  x-axis, 
and  which  passes  through  the  origin  and  also  through  the  point  (2,  3). 

6.  Find  the  points  of  intersection  of  the  tw^o  circles 

a;2  ^  ^2  _  4  3,  _  2  ?/  _  31  =  0    and   x2  +  ?/2_4x  +  2?/  +  l=0. 

7.  Circles  are  drawn  having  th  ir  centers  at  the  vertices  of  the 
triangle  (7,  2),  (-1,  -4)  and  (3,  3),  respectively,  and  each  23assing  through 
the  center  of  a  fourth  circle  which  circumscribes  this  triangle  ;  find  their 
equations,  their  common  chords,  and  their  radical  center. 

8.  Circles  having  the  sides  of  the  triangle  (7,  2),  (-1,  -4),  (3,  3)  as 
diameters  are  drawn ;  find  their  equations,  their  radical  axes,  and  their 
radical  center. 


166  ANALYTIC   GEOMETRY  [Ch.  VII. 

9.    Find  the  equation  of  the  circle  passing  through  the  origin  and 
the  point  {x-^,  y-^),  and  having  its  center  on  the  2/-axis. 

10.  The  point  (3,  "5)  bisects  a  chord  of  the  circle  x'^  -\-  ]f-  —  277 ;  find 
the  equation  of  thab  chord. 

11.  A  circle  touches  the  line  4x  +  3?/  +  3  =  0  at  the  point  (~3,  3) 
and  passes  through  the  point  (5,  9)  ;  find  its  equation. 

12.  A  circle,  whose  center  coincides  with  the  origin,  touches  the  line 
7a;  —  ll?/  +  2  =  0;  find  its  equation. 

13.  At  the  points  in  which  the  circle  x'^  +  y'^  —  ax  —  hy  =  0  cuts  the 
axes,  tangents  are  drawn ;  find  the  equations  of  these  tangents. 

14.  A  circle,  whose  radius  is  7,  touches  the  line  Dy  =  lx  —  1  at  the 
point  (8,  11)  ;  find  the  equation  of  this  circle. 

15.  A  circle  is  inscribed  in  the  triangle  (7,  2),  (—1,  "•!),  (3,  3);  find 
its  equation ;  find  also  the  equations  of  the  polars  of  the  three  vertices 
with  regard  to  this  circle. 

16.  Through  a  fixed  point  (Xy,  y^)  a  secant  line  is  drawn  to  the  circle 
^2  _|.  ^2  _  ^2 .  -^Yi^  the  locus  of  the  middle  point  of  the  chord  which  the 
circle  cuts  from  this  secant  line,  as  the  secant  revolves  about  the  given 
fixed  point  (x^,  y-^) . 

17.  Prove  analytically  that  an  angle  inscribed  in  a  semicircle  is  a 
right  angle. 

18.  Prove  analytically  that  a  radius  drawn  perpendicular  to  a  chord 
of  a  circle  bisects  that  chord. 

19.  Show  that  the  distances  of  two  points  from  the  center  of  a  circle 
are  proportional  to  the  distances  of  each  from  the  polar  of  the  other. 

20.  Two  straight  lines  touch  the  circle  x^  -\-  y'^  —  5  x  —  3  y  -\-  Q  =  0, 
one  at  the  point  (1, 1)  and  the  other  at  the  point  {2,  3)  ;  find  the  pole 
of  the  chord  of  contact  of  these  tangents. 

21.  Find  the  condition  among  the  coefficients  that  must  be  satisfied 
if  the  circles 

x'i  +  y'i  +  2  GyX  +  2  F^y  =0     and     x"^  +  y'^  +  2  G.^x  +  2F^y  =  Q 
shall  touch  each  other  at  the  origin. 

22.  Determine  G,  F,  and  C  so  that  the  circle 

x^-  +  y'^  +  2Gx-\-2Fy^  C  =  0 
shall  cut  each  of  the  circles 

a;2^^2_4a;_2y  +  4  =  0     and     x^ -h  y-  +  4:X  +  2  y  =  1 
at  right  angles  (cf.  Art.  100). 


100.]  THE  CIRCLE  167 

23.  Given  the  two  circles 

a:2  +  ?/2-4x-2?/  +  4=0     and     x^  ■}-  y'^  -\-  ^x  +2  y  -  ^  =  0'^ 
find  the  equation  of  their  common  tangents. 

24.  Find  the  radical  axis  of  the  circles  in  example  23 ;  show  that  it 
is  perpendicular  to  the  line  joining  the  centers  of  the  given  circles,  and 
find  the  ratio  of  the  lengths  of  the  segments  into  which  the  radical  axis 
divides  the  line  joining  the  centers.  How  is  this  ratio  related  to  the 
radii  of  the  circles  ?  Is  this  relation  true  for  any  pair  of  circles  what- 
ever ? 

25.  Given  the  three  circles  : 

a;2  +  /  -  16  a:  +  60  =  0,  3  x^  +  3  ?/2  _  36  a;  +  81  =  0, 

and  a;2  +  ?/2  _  16  a;  -  12  ?/  +  84  =  0 ; 

find  the  point  from  which  tangents  drawn  to  these  three  circles  are  of 
equal  length,  also  find  that  length.  How  is  this  point  related  in  position 
to  the  radical  center  of  the  given  circles  ?  Prove  that  this  relation  is  the 
same  for  any  three  circles. 

26.  Find  the  locus  of  a  point  which  moves  so  that  the  length  of  the 
tangent,  drawn  from  it  to  a  fixed  circle,  is  in  a  constant  ratio  to  the  dis- 
tance of  the  moving  point  from  a  given  fixed  point. 

27.  Let  P  be  a  fixed  point  on  a  given  circle,  T  a  point  moving  along 
the  circle,  and  Q  the  point  of  intersection  of  the  tangent  at  T  with  the 
perpendicular  upon  it  from  P ;  find  the  locus  of  Q. 

Suggestion.  Use  polar  coordinates,  P  being  the  pole,  and  the  diam- 
eter through  P  the  initial  line. 

28.  Find  the  length  of  the  common  chord  of  the  two  circles 

(x  —  rt)2  +  (?/  —  bY  =  T^     and     {x  —  6)^  +  (^  —  «)^  =  ^^• 
From  this  find  the  condition  that  these  circles  shall  touch  each  other. 

29.  If  the  axes  are  inclined  at  60°,  prove  that  the  equation 

x'^  -\-  xy  -\-  y"^  —  ^:X  —^  y  —  2  —  ^ 
represents  a  circle ;  find  its  radius  and  center. 

30.  What  is  the  obliquity  of  the  axes  if  the  equation 

a;2  +  V3  a:2/-f?/"2  —  4a;  —  63/-|-5  =  0 
represents  a  circle?     AVhat  is  its  radius? 

31.  For  what  point  on  the  circle  a:^  +  ?/2  _  9  ^^-e  the  subtangent  and 
the  subnormal  of  equal  length?  the  tangent  and  normal?  the  tangent 
and  subtangent  ? 


108  ANALTTJC   GEOMETRY  [Ch.  VH. 

32.  An  equilateral  triangle  is  inscribed  in  the  circle  a:^  +  ?/2  —  4  with 
its  base  parallel  to  the  a:-axis ;  through  its  vertices  tangents  to  the  circle 
are  drawn,  thus  forming  a  circumscribed  triangle ;  find  the  equations, 
and  the  lengths,  of  the  sides  of  each  triangle. 

33.  The  poles  of  the  sides  of  each  triangle  in  example  32  are  the 
vertices  of  a  triangle ;  find  the  equations  of  its  sides,  and  draw  the  figure. 

34.  A  chord  of  the  circle  a;^  +  ?/2  —  22  a:  —  4  !/  +  25  =  0  is  of  length 
4  V5,  and  is  parallel  to  the  line  2x  +  ^  +  7  =  0;  find  the  equation  of  the 
chord,  and  of  the  normals  at  its  extremities. 

35.  Find  the  equation  of  a  circle  through  the  intersection  of  the 
circles  ^2+  ?/2-4  =  0,  x^-\-y'^  —  '2.x— ^y+b  =  0,  and  tangent  to  the  line 

x  +  y-o  =  i). 

36.  The  length  of  a  tangent,  from  a  moving  point,  to  the  circle 
a;2  +  ^2=6  is  always  twice  the  length  of  the  tangent  from  the  same  point 
to  the  circle  a:-  +  3/^  +  3  (a:  +  ?/)  =  0.  Find  the  equation  of  the  locus  of 
the  moving  point. 

37.  Find  the  locus  of  the  vertex  of  a  triangle  having  given  the  base 
=  2  «,  and  the  sum  of  the  squares  of  its  sides  =  2  J^. 

38.  Find  the  locus  of  the  middle  points  of  chords  drawn  through  a 
fixed  point  on  the  circle  x^  +  3/'^  =  a^. 

39.  Through  the  external  point  P^  =  (xj,  y^),  a  line  is  drawn  meeting 
the  circle  x'^  +  y'^  =  a'^  in  Q  and  R;  find  the  locus  of  middle  point  of  P^Q 
as  this  line  revolves  about  Py 

40.  A  point  moves  so  that  its  distance  from  the  point  (1,  3)  is  to  its 
distance  from  the  point  (~4,  1)  in  the  ratio  2:3.  Find  the  equation 
of  its  locus. 

41.  Do  the  circles 

4a;2  +  4?/2  +  4x-12?/  +  l=0     and     2x^-{-2y^  +  y  =  0 
intersect?     Show  in  two  ways. 

42.  Find  the  equation  of  a  circle  of  radius  VSS  which  passes  through 
the  points  (2,  1)  and  ("3,  4). 

43.  What  are  the  equations  of  the  tangent  and  the  normal  to  the 
circle  x'^ -{- y'^  =  13,  —  these  lines  passing  through  the  point  (2,-3)? 
through  the  point  (0,  6)  ? 

44.  Find  the  equations  of  the  tangents  through  (2,  3)  to  the  circle 

9(a;2+  y^)+Qx-12y  +  4.  =  0. 


100.]  THE  CIRCLE  169 

45.  At  what  angle  do  the  ch-cles  ar^  +  ^/^  +  Oa;  —  2y  +  5  =  0  and 
x^  -\-  y'^  -\-  ^  X  -\- 2  y  —  5  =  0  intersect  each  other  ? 

46.  A  diameter  of  the  circle  4a;2  +  4?/-  +  8x  —  12y  +  l  =  0  passes 
through  the  point  (1,  -1).  Find  its  equation,  and  the  equation  of 
the  chords  which  it  bisects. 

47.  Find  the  locus  of  a  point  such  that  tangents  from  it  to  two  con- 
centric circles  are  inversely  proportional  to  the  radii  of  the  circles. 

48.  Find  the  locus  of  a  point  which  moves  so  that  its  distances  from 
two  fixed  points  are  in  constant  ratio  k.  Discuss  the  locus  and  draw 
the  figure. 

49.  A  point  moves  so  that  the  square  of  its  distance  from  the  base 
of  an  isosceles  triangle  is  equal  to  the  product  of  its  distances  from  the 
other  two  sides.     Show  that  the  locus  is  a  circle. 

50.  Prove  that  the  two  circles 

x2  +  2/2  +  2  G^x  +  '2F^y^C^  =  0  and  x^ -^  y'^ -\- 2  G^x -\- 2  F,y  +  C^  =  0 

are  concentric  if  (r^  =  G^  and  F^  =  F^]  that  they  are  tangent  to  each 
other  if 

V(G,  -  G,y  +  (F,  -  F,y  =VG,^  +  F,'  -c,±VG,^  +  i'V  -  ^2 ; 

and  find  the  condition  among  the  constants  that  these  cu'cles  intersect 
orthogonally,  i.e.,  at  right  angles  to  each  other. 


CHAPTER   VIII 
THE  CONIC  SECTIONS 

101.  In  Art.  48,  which  should  now  be  carefully  re-read, 
a  conic  section  was  defined ;  its  general  equation  was  de- 
rived; its  three  species,  viz.,  the  parabola,  ellipse,  and  hyper- 
bola, were  mentioned  ;  and  a  brief  discussion  of  the  nature 
and  forms  of  the  curve  was  given.  In  the  present  chap- 
ter, each  of  these  three  species  will  be  examined  somewhat 
more  closely  than  was  done  in  Chapter  IV,  and  some  general 
tlieorems  concerning  its  tangents,  normals,  diameters,  chords 
of  contact,  and  polars  will  be  proved. 

The  general  equation  (Art.  48)  of  the  conic  section 
might  here  be  assumed,  and  the  special  forms  for  the  parab- 
ola, the  ellipse,  and  the  hyperbola  be  derived  from  it ;  but, 
partly  as  an  exercise,  and  partly  for  the  sake  of  freedom 
to  choose  the  axes  in  the  most  advantageous  ways,  the  equa- 
tions will  here  be  re-derived,  as  they  are  needed,  from  the 
definitions  of  the  curves. 

I.    THE   PAKABOLA 
Special  Equation  of  Second  Degree 

Ax^  +  2Goc  +  2Ftj+  C  =  0,  or  Bij^  -{-2Chc  +  2Fy  +  C  =  0 

102.  The  parabola  defined.  A  parabola  is  the  locus  of 
a  point  which  moves  so  that  its  distance  from  a  fixed  point, 
called  the  focus,  is  equal  to  its  distance  from  a  fixed  line 

170 


Ch.  VIII.  101-103.]         THE   CONIC   SECTIONS  171 

called  the  directrix.     It  is  the  conic  section  with  eccentricity 
e  =  1  (cf.  Art.  48). 

The   equation   of  a  parabola,  with   any  given  focus   and 
directrix,  can  be  obtained  directly  from  this  definition. 

Example.     To  find  the  equation  of   the  parabola  whose   directrix 
is  the  line  x-2y-l=0^  and  whose  focus  is  the  point  (2,  -3). 
Let  P  =  (x,  y)  be  any  point  on  the  parabola(see  Fig.  79)  ; 

then  "^-^ is  the  distance  of  P  from  the  directrix  (Art.  64), 

+  Vs 


and  V(x  -  2)^  +  (y  +  3)2  is  the  distance  of  P  from  the  focus  (Art.  26); 
x-2y-l  _ 


hence  — — ^-= —  -  V(a;  -  2)2  +  {y  +  3)2,   by  definition ; 

that  is,  4x2  +  4a:2/ +^2_i8^_f,962/  + 64  =  0; 

which  is  the  required  equation. 

The  equation  obtained  in  this  way  is  not,  however,  in  the 
most  suitable  form  from  which  to  study  the  properties  of  the 
curve,  but  can  be  simplified  by  a  proper  choice  of  axes. 
In  Art.  48  it  was  shown  that  the  parabola  is  symmetrical 
witli  respect  to  the  straight  line  through  the  focus  and  per- 
pendicular to  the  directrix,  and  that  it  cuts  this  line  in  only 
one  point.  If  this  line  of  symmetry  is  taken  as  the  2:-axis, 
the  equation  will  have  no  ^-term  of  first  degree  [cf.  Art.  48, 
eq.  (3)] ;  while  if  the  point  of  intersection  of  the  curve  with 
this  axis  be  taken  as  origin,  the  equation  will  have  no  con- 
stant term,  since  the  point  (0,  0)  must  satisfy  the  equation. 
With  this  choice  of  axes,  the  equation  of  the  parabola  will 
reduce  to  a  simple  form,  which  is  usually  called  the  first 
standard  equation  of  the  parabola. 

103.  First  standard  form  of  the  equation  of  the  parabola. 
Let  D'B  be  the  directrix  of  the  parabola,  and  F  its  focus ; 


172  ANALYTIC  GEOMETRY  [Ch.  VIII. 

also  let  the  line  ZFX,  perpendicular 
to  the  directrix,  be  the  a:-axis ;  denote 
the  fixed  distance  ZF  by  2  p,  and  let 
^  0,  its  middle  point,  be  the  origin  of 
coordinates;  then  the  line  OZ,  per- 
pendicular to  OX,  is  the  ^-axis.  Let 
P  =  (^x,  y)  be  any  point  on  the  curve, 
Fig, 76.^  and  draw  liQP  perpendicular  to  OY^ 

also  draw  the  ordinate  MP^  and  the 
line  FP.     The  line  FP  is  called  the  focal  radius  of  P. 

Then  ZO=OF  =  p, 

and  the  equation  of  the  directrix  \^  x+p  =  0,  .   .   .   (1) 

while  the  focus  is  the  point  (jo,  0).    .      .      .     (2) 

Again,  from  the  definition  of  the  parabola, 
FP  =  LP\    [geometric  equation] 


but      FP  =  ^(x-py^y\  and  LP=ZO+  OM=p  +  x  ; 
hence  V(a;  -  pj^  +  y'^  =  (x  +^), 

whence  y^  =  ^p^c,  .         .         .         [41] 

which  is  the  desired  equation. 

This  first  standard  form  [41]  is  the  simplest  equation  of 
the  parabola,  and  the  one  which  will  be  most  used  in  the 
subsequent  study  of  the  curve.  It  will  be  seen  later 
(Chapter  XII)  that  any  equation  which  represents  a  parab- 
ola can  be  reduced  to  this  form. 

104.  To  trace  the  parabola  y^  =  'i.px.  From  equation 
[11]  it  follows  : 

(1)  That  the  parabola  passes  through  the  point  0,  half 
way  from  the  directrix  to  the  focus.  This  point  is  called 
the  vertex  of  the  curve. 

(2)  That  the  parabola  is  symmetrical  with  regard  to  the 


103-106.]  THE  CONIC  SECTIONS  173 

a;-axis  ;  ^.e.,  with  regard  to  the  line  through  the  focus  per- 
pendicular to  the  directrix ;  this  line  is  called  the  axis  *  of 
the  curve. 

(3)  That  X  has  always  the  same  sign  as  the  constant  j?, 
^.e.,  that  the  entire  curve  and  its  focus  lie  on  the  same  side 
of  a  line  parallel  to  the  directrix,  and  micfway  between  the 
directrix  and  the  focus.  ' 

(4)  That  X  may  vary  in  magnitude  from  0  to  oo,  and  when 
X  increases,  so  also  does  y  (numerically) ;  hence  the  parabola 
is  an  open  curve,  receding  indefinitely  from  its  directrix  and 
its  axis. 

The  parabola  is  then  an  open  curve  of  one  branch  which 
lies  on  the  same  side  of  the  directrix  as  does  the  focus  ; 
when  constructed  it  has  the  form  shown  in  Fig.   76. 

105.  Latus  rectum.  The  chord  through  the  focus  of  a 
conic,  parallel  to  the  directrix,  is  called  its  latus  rectum.  In 
the  figure  this  chord  is  R^R. 

Now  WR  =  2FR  =  'lSR=1ZF=4:p. 

Hence  the  length  of  the  latus  rectum  of  the  parabola  is  4:p; 
that  is,  it  is  equal  to  the  coefficient  of  x  in  the  first  standard 
equation. 

106.  Geometric  property  of  the  parabola.  Second  standard 
equation.  Equation  [41]  may  be  interpreted  as  stating 
an  intrinsic  property  of  the  parabola,  —  a  property  which 
belongs  to  every  point  of  the  parabola,  whatever  coordinate 
axes  be  chosen.  For  (see  Fig.  76)  the  equation  y^  =  4:px 
gives  the  geometric  relation 

or,  expressed  in  words, 

*  The  axis  of  a  curve  should  be  carefully  distinguished  from  an  axis  of 
coordinates;  though  they  often  are  coincident  lines  in  the  figures  to  be 
studied. 


174  ANALYTIC  GEOMETRY  [Ch,  VIII. 

If  from  any  point  on  the  parabola^  a  perpendicular  is  drawn 
to  the  axis  of  the  curve^  the  square  on  this  perpendicular  is 
equivalent  to  the  rectangle  formed  by  the  latus  rectum  and  the 
line  from  the  vertex  to  the  foot  of  the  perpendicular. 

This  geometric  property  enables  one  to  write  down  immedi- 
ately the  equation  of  the  parabola,  whenever  the  axis  of 
the  curve  is  parallel  to  one  of  the  coordinate  axes. 

E.g.^  if  the  vertex  of  the  parabola  is  the  point  A  =  (li^  ^), 
and  its  axis  is  parallel  to  the  a^-axis,  as  in  tlie  figure,  let 

F  be  the  focus  and  P  =  (a;,  y) 
be  any  point  on  the  parabola  ; 
draw  MP  perpendicular  to  the 
axis  AIC.     Then 

MP^  =  4:AF-AM, 

i.e.,     (y-k')^  =  4=2^(00-71),  .  [42] 

which  is  the  equivalent  algebraic 
equation.  This  may  be  taken  as 
a  second  standard  form  of  the  equation,  representing  the 
parabola  with  vertex  at  the  point  (A,  k'),  with  axis  parallel 
to  the  a?-axis,  and,  if  p  is  positive,  lying  wholly  on  the  posi- 
tive side  of  the  line  x=  h. 

Equation  [42]  evidently  may  be  reduced  to  equation  [41] 
by  a  transformation  of  coordinates  to  parallel  axes  through 
the  vertex  (A,  ^),  as  the  new  origin. 

Again,  suppose  the  position  of  the  parabola  to  be  that 
represented  in  Fig.  78.  The  vertex  is  ^  =  (A,  A:),  and  the 
axis  of  the  parabola  is  parallel  to  the  ?/-axis.  Let  P  =  (x,  y') 
be  any  point  on  the  curve,  and  draw  MP  perpendicular  to 
the  axis  of  the  curve. 


Then  MP  =  4  AF  •  AM  [geometric  property  J 

—  4  (^—p^AM.,     [since  AF  is  negative] 


106-107.] 


THE  CONIC  SECTIONS 


175 


whence,  substituting  the  coordinates  of  A  and  P, 

{x-hf  =  -^p{y--k),  .  .  .  [43] 
which  is  another  form  for  the  second  standard  equation  of 
the  parabola. 


Fig.  78. 


EXERCISES 

Construct  the  following  parabolas,  and  find  their  equations  : 

1.  having  the  focus  at  the  point  (~1,  3),  and  for  directrix  the  line 
3ar-5?/  =  2  (cf.  Art.  102); 

2.  having  the  focus  at  the  origin,  and  for  directrix  the  line 

.    2x-y  +  ^  =  0', 

3.  with  the  vertex  at  the  origin,  and  the  focus  at  the  point  (3,  0); 

4.  with  the  vertex  at  the  origin,  and  the  focus  at  the  point  (0,  "3) ; 

5.  with  the  vertex  at  the  point  (~2,  5),  and  the  focus  at  the  point 

(-2, 1); 

6.  with  the  vertex  at  the  point  (~2,  -4),  and  the  focus  at  the 
point  (1,  -4) ; 

7.  having  the  focus  at  the  point  (2p,  0),  and  for  directrix  the  line 
x  =  Q. 

8.  What  is  the  latus  rectum  of  each  of  the  parabolas  of  exercises  3  to  6. 

9.  Describe  the  effect  produced  on  the  form  of  a  parabola  by  increas- 
ing or  decreasing  the  length  of  its  latus  rectum. 

107.  Every  equation  of  the  form  Ax'^  +  2  Goc  +  2  Fy  +  C  =  O, 
or  By'i  -^  2  Gjc  +  2  Fy  +  C  =  Oy  represents  a  parabola  whose 
axis  is  parallel  to  one  of  the  coordinate  axes. 

Equations  [41],  [42],  and  [43]  are  of  the  form 


176  ANALYTIC  GEOMETRY  [Ch.  VIII. 

that  is,  each  has  one  and  only  one  term  containing  the 
square  of  a  variable,  and  no  term  containing  the  product 
of  the  two  variables.  Conversely,  it  may  be  shown  that 
an  equation  of  either  of  these  forms  represents  a  parabola 
whose  axis  is  parallel  to  one  of  the  coordinate  axes. 

A  numerical  example  will  first  be  discussed,  by  the 
method  which  has  already  been  employed  in  connection 
with  the  equation  of  the  circle  (Art.  79),  and  which  is 
applicable  also  in  the  case  of  the  other  conies.  It  is  the 
method  of  reducing  the  given  equation  to  a  standard  form, 
and  is  analogous  to  "completing  the  square"  in  the  solu- 
tion of  quadratic  equations. 

Example.     Given  the  equation 

25^2  -  30^  -  50a;  +  89  =  0, 
to  show  that  it  represents  a  parabola ;  and  to  find  its  vertex,  focus,  and 
directrix. 

Divide  both  members  of  the  equation  by  25,  and  complete  the  square 
of  the  ^-terms ;  the  equation  may  then  be  written 

that  is,  (2/-|)2  =  2(a:-f), 

whence  (2/  -  f  )^  =  4  •  i  •  (.r  -  f ) . 

Now  this  equation  is  in  the  second  standard  form  (cf.  equation  [42]), 
and  therefore  every  point  on  its  locus  has  the  geometric  property  given 
in  Art.  106;  and  the  locus  is  a  parabola.  The  vertex  is  at  the  point 
(f ,  I)  ;  its  axis  is  parallel  to  the  x-axis,  extending  in  the  positive  direc- 
tion ;  and,  since  p  =  1,  its  focus  is  at  the  point  (f^,  |),  and  the  directrix 
is  the  line  x  =  \^. 

Consider  now  the  general  equation,  and  apply  the  same 
method,  taking  for  example  the  second  form,  viz.  : 

Aa^+^ax  +  'lFy^-  (7=0. 
Dividing  both  numbers  of  the  equation  by  A,  completing  the 
square  of  the  a;-terms,  and  transposing,  the  equation  becomes 


107-108.]  THE  CONIC  SECTIONS  177 

\       aJ            A\f  2AF 

whence  (^+~r)  —  ^( )\^~       ~~ 


AJ         \     2AJV  2AF 

Comparing  this  equation  with  the  standard  equation  [43], 
it  is  seen  that  its  locus  is  a  parabola,  whose  axis  is  parallel 
to  the  y-axis,  extending  in  the  negative  direction  if  A  and  IB 
have  like  signs,  and  in  the  positive  direction  if  A  and  F  have 

unlike  signs.     Its  vertex  is  at  the  point  I  — -,  — 

XT 

and,  since  p  =  —  — — ,  its  focus  is  at  the  point 


( 


and  its  directrix  is  the  line    y  = 


a  a^-F^-  AC 

A'  2AF 


2AF 


Note.  The  transformation  just  given  fails  if  A  =  0  ot  ii  F  =  0,  for 
in  that  case  some  of  the  terms  in  the  last  equation  are  infinite.  If,  how- 
ever, A=0,  the  given  equation  becomes  2  Gx  +  2  Fy  +  e  =  0 ;  and,  this 
being  of  the  first  degree,  represents  a  straight  line.  If,  on  the  other 
hand,  F=0,  the  given  equation  reduces  to  Ax'^  -{-  Gx  +  C  =  0,  and  repre- 
sents two  straight  lines  each  parallel  to  the  ?/-axis  ;  they  are  real  and 
distinct,  real  and  coincident,  or  imaginary,  depending  upon  the  value  of 
G^  —  AC.  These  lines  may  be  regarded  as  limiting  forms  of  the  parab- 
ola (see  Chapter  XII). 

EXERCISES 

Determine  the  vertex,  focus,  latus  rectum,  equation  of  the  directrix 
and  of  the  axis  for  each  of  the  following  parabolas ;  also  sketch  each 
of  the  figures : 

1,  i/  -  5x  +  4:y  -10  =0]  3.    5?/ -  1  =  3?/2  +  4x; 

2.  3  a;2  +  12  X  +  4  ?/  -  8  =  0 ;  4.   y^-  -\- 2  y  -  12  x  -  U  =  0. 

108.  Reduction  of  the  equation  of  a  parabola  to  a  standard  form.  In 
Art.   102   it  was  shown   that  the  equation  of   a  parabola   having  any 

TAN.    AX.    GEOM.  —  12 


178 


ANALYTIC  GEOMETBT 


[Ch.  VIII. 


Fig. 79. 


given  directrix  and  focus  is  in  general  not  as  simple  as  the  standard  equa- 
tion.    It  will  now  be  shown  that  if  the  coordinate  axes  be  transformed 

so  as  to  be  parallel  to  the  axis  and 
directrix  of  the  curve,  the  equation  wiU 
be  reduced  to  a  standard  form.  For  ex- 
ample, the  equation  of  the  parabola  with 
focus  at  (2,  -3),  and  having  for  directrix 
the  line  x  —  2y  — 1  =  0,  was  found  to  be 
4  a;2  +  4  a:^/  +  2/2  -  18  a:  +  26  ?/  +  64  =  0. 

The  axis  of  the  curve  is  a  line  through 
(2,  ~3)  and  perpendicular  to 

x-2?/-  1  =  0; 

its  equation  is  2  x  +  y  =  1,  and  it  cuts 

the  a:-axis  at  the  angle  0  =  tan-i(-2). 

The  point  Z  is  the  intersection  of  the  directrix  and  axis,  and  may  be 

found  from  the  two  linear  equations  representing  these  lines  ;  the  vertex 

A  is  the  point  bisecting  ZF.     If,  then,  the  axes  are  rotated  through 

the  angle  ^  =  tan-i(-2),  the  equation  will  be  reduced  to  the   second 

standard  form,  [42] ;  and  if  the  origin  be  also  removed  to  the  vertex 

A,  the  equation  will  be  fiu'ther  reduced  to  the  first  standard  form,  [41]. 

7 
The  point  Z  is  (|,  -\),  A  is  (If,  -f) ;  hence,  j9  =  .4F  =  -^:,  and  trans- 

forming  the  axes  through  the  angle  ^  =  tan-i(-^2),  to  the  new  origin 

9g 

(|3^-|)^  the  equation  of  the  parabola  reduces  to  y'^  =-^a:. 

V5 
The  problem  of  reducing  any  equation  representing  a  parabola  to  its 
standard  form  is  taken  up  more  fully  in  Chap.  XII. 


EXERCrSES 

Find,  and  reduce  to  the  first  standard  form,  the  equation  of  each  of 
the  following  parabolas ;  also  make  a  sketch  of  each  figure  : 

1.  with  focus  at  the  point  (-1,  3),  and  having  for  directrix  the  line 

2.  with  focus  at  the  point  (-8,  -|),  and  having  for  directrix  the  line 

2a;  +  7?/-8  =  0; 

3.  with  focus  at  the  point  (a,  h),  and  having  for  directrix  the  line 

a     b 


108-109.]  THE  CONIC  SECTIONS  179 

II.    THE   ELLIPSE 

Special  Equation  of  the  Second  Degree 

Aqc^  +  U2/2  +  2  6?a5  +  ^Fy  +  C  =  0 

109.  The  ellipse  defined.  An  ellipse  is  the  locus  of  a 
point  which  moves  so  that  the  ratio  of  its  distance  from 
a  fixed  point,  called  the  focus,  to  its  distance  from  a  fixed 
line,  called  the  directrix,  is  constant  and  less  than  unity. 
The  constant  ratio  is  called  the  eccentricity  of  the  ellipse. 
This  curve  is  the  conic  section  with  eccentricity  e<\. 
(cf.  Art.  48.) 

The  equation  of  an  ellipse  with  any  given  focus,  directrix, 
and  eccentricity  may  be  readily  obtained  from  this  definition. 

Example.  An  ellipse  of  eccentricity  ^  has  its  focus  at  (2,  -1),  and 
has  the  line  x  +  2y  ■=  d  for  directrix.  Let  P=(x,  y)  (Fig.  85)  be  any 
point  on  the  curve,  i^the  focus,  and  PQ  the  perpendicular  from  P  to 
the  directrix. 

Then  FP  =  ^QP] 

but  FP  =  V(a;  -  2)2  +  {y  +  l)^,    QP  =  x+^y-^    (Arts.  26,  64), 

+  V5 

hence  (x  -  2y  +  {y  +  ly  =  ^  (x -\- 2y  -  by-, 

that  is,  41  x^  -Wxy  +  29  zf  -  140  a;  +  170 y  +  125  =  0; 

which  is  the  equation  of  the  given  ellipse. 

As  in  the  case  of  the  parabola,  so  also  here,  a  particular 
choice  of  the  coordinate  axes  gives  a  simpler  form  for  the 
equation  of  the  ellipse ;  an  equation  which  is  more  suitable 
for  the  study  of  the  curve,  and  to  which  every  equation 
representing  an  ellipse  can  be  reduced.  As  has  been  seen  in 
Art.  48,  the  curve  is  symmetrical  with  respect  to  the  line 
through  the  focus  and  perpendicular  to  the  directrix ;  and 
cuts  that  line  in  two  points,  one  on  either  side  of  the  focus. 
The  equation  of  the  ellipse  will  be  in  a  simpler  form  if  this 


180 


ANALYTIC  GEOMETRY 


[Ch.  VIII. 


line  of  symmetry  is  chosen  as  the  rr-axis,  with  the  origin  half 
way  between  its  two  points  of  intersection  with  the  curve. 
The  resulting  equation  is  the  first  standard  form  of  the  equa- 
tion of  the  ellipse. 

110.   The  first  standard  equation  of  the  ellipse.     Let  F  be  the 
focus,  D'D  the  directrix,  and  ZFX  the  perpendicular  to  BD' 

through  i^,  cutting 
the  curve  in  the  two 
points  A'  and  A 
(Art.  48)*.  Denote 
by  2  a  the  length 
of  AA\  and  let  0 
be  its  middle  point, 
so  that 

A0=  OA'  =  a. 

Let   ZX  be    the  2:-axis,    0  the  origin,   and    OY,  perpen- 
dicular to  OX,  the  y-axis.     Then,  by  the  definition  of  the 

ellipse, 

AF=  eZA,     and     FA'  =  eZA'  ; 

.  •.       AF+FA'  =  e  {ZA  -f  ZM)  =  e{ZA  +  ZA  +  AA'}, 

i.e.,  AA'  =  e(2ZA+AA'), 

whence  2  a  =  2<ZA  +  ^ 0)  =  2  eZO  ; 

therefore       ZO  =  -, 


D 

Y 

B 

L 

^""^v^ 

'''  '  '\. 

Z 

^  ( 

0                  \ 

^  " 

A 

Y a-e- • 

M 

r 

D' 

Fig 

B' 

.80. 

and  the  equation  of  the  directrix  z's  a;  -f-  -  =  0. 

e 

Again,  FA'-AF=e  {ZA'  -  ZA} ; 

i.e.,  FO  -f  OA'  -  (AO  -  FO}  =  eAM, 

whence  2  FO  =2ae\ 


(1) 


*  Thi3  equation  may  also  be  easily  derived  independently  of  Art.  48, 
cf.  Arts.  103,  116. 


109-110.]  THE   CONIC   SECTIONS  181 

therefore  FO  =  ae, 

and  t\\Q  focus  F  is  the  point  (—  ae,  0).       .  .  .  (2) 

Now,  for  any  point  P  on  the  curve,  draw  the  ordinate  MP 
and  the  perpendicular  LP  to  the  directrix  ;   then 

FP  =  eLP^     [geometric  equation]   .      o     .      (3) 

but   FP  =  ^{x^aey^-y\  LP=-  +  xi 

e 

hence  {ae  +  xf  -V  y"  =  e^  (^+  ~T'      *      *      *     -  (^) 

that  is,  (1  -  e^^x^  -\- y'^  =  a?  (1  -  e^),    ...      (5) 

tliatis,  ^+    ,y    2^  =  1 (6) 

From  equation  (6),  the  intercepts  of  the  curve  on  the  ^-axis 
are  ±  a  Vl  —  e^.  Both  intercepts  are  real,  since  e<l;  hence 
the  ellipse  cuts  the  ?/-axis  in  two  real  points,  B  and  B',  on 
opposite  sides  of  the  origin  0  and  equidistant  from  it.  If 
OB  is  denoted  by   +  ^,  so  tliat 

¥  =  a\l-e'),       ...         (7) 

equation  (3)  takes  the  form 

This  is  the  simplest  equation  of  the  ellipse,  and  will  be 
most  used  in  the  subsequent  study  of  the  properties  of  that 
curve.  As  will  be  seen  in  Chapter  XII,  every  equation 
representing  an  ellipse  can  be  reduced  to  this  form. 

*  It  a  =  h  (i.e.,  if  e  =  0)  this  equation  represents  a  circle.  The  ellipse, 
then,  includes  the  circle  as  a  special  case.  Tn  other  words  :  a  circle  is  an 
ellipse  whose  eccentricity  is  zero. 


182 


ANALYTIC   GEOMETRY 


[Ch.  YIII. 


QC^ 


y''  - 


111.   To  trace  the  ellipse  ^  +  t^=1« 


From  equation  [44] 

it  follows  that ; 

(1)  The  ellipse  is  symmetrical  with  regard  to  the  a:-axis  ; 
^.e.,  with  regard  to  the  line  through  the  focus  and  perpen- 
dicular to  the  directrix ;  this  line  is  therefore  called  the 
principal  axis  of  the  curve ; 

(2)  The  ellipse  is  symmetrical  with  regard  to  the  i/-axis 
also  ;  i.e.^  with  regard  to  a  line  parallel  to  the  directrix  and 
passing  through  the  mid-point  of  the  segment  AA'  (Fig.  81) 
which  the  curve  cuts  from  its  principal  axis  ; 

(3)  For  every  value  of  x  from  —a  to  +a,  the  two  cor- 
responding values  of  y  are  real,  equal  numerically,  but 
opposite  in  sign  ;  and  for  every  value  of  y  from  —  5  to  5, 
the  two  values  of  x  are  real  and  equal  numerically,  but 
opposite  in  sign  ;  and  that  neither  x  nor  y  can  have  real 
values  beyond  these  limits. 

The  ellipse  is,  therefore,  a  closed  curve,  of  one  branch, 
which  lies  wholly  on  the  same  side  of  the  directrix  as  the 
focus  ;  and  the  curve  has  the  form  represented  in  Fig.  80, 
—  which  agrees  with  the  foot-note  on  p.  71. 


D 

Y 
B 

D, 

L 

"^ 

^ 

L' 

,^- 

0 

1  \      > 

\ 

Z 

Z'   X 

M      "f^ ci-e- 

\     1 
\  1 

._._ 

M 

J 

h' 

y 

D' 

Fig 

B' 
.81. 

D', 

The  segment  AA'  (Fig.  81)  of  the  principal  axis  inter- 
cepted by  the  curve  is  called  its  major  or  transverse  axis ; 


111-112.]  THE  CONIC   SECTIONS  183 

the  corresponding  segment  B' B  is  its  minor  or  conjugate  axis. 
From  the  symmetry  of  the  curve  with  respect  to  these  axes 
it  follows  that  it  is  also  symmetrical  with  respect  to  their 
intersection  0,  the  center  of  the  ellipse.  It  follows  also  that 
the  ellipse  has  a  second  focus  at  F'  =  (ae^  0)  C^^^ig-  81)  and 

a  second  directrix  J>'ii>i  —  the  line  x —  =0  —  on  the  posi- 

e 

tive  side  of  the  minor  axis,  and  symmetrical  to  the  original 
focus  and  directrix,  respectively.* 

The  latus  rectum  of  the  ellipse,  i.e.,  the  focal  chord  parallel 
to  the  directrix  (Art.  105),  is  evidently  twice  the  ordinate 
of  the  point  whose  abscissa  is  ae. 

But  if   Xi  =  ae^    Vi  —  ^  "^^1  —  ^^ ;    or,  since    h  =  a  VI  —  e^ 

y^  —  —  .     Hence  the  latus  rectum  is . 

a  a 

112.  Intrinsic  property  of  the  ellipse.  Second  standard 
equation.  Equation  [-14]  states  a  geometric  property  which 
belongs  to  every  point  of  the  ellipse,  whatever  the  coordi- 
nate axes  chosen,  and  to  no  other  point  :  viz.,  if  P  be  any 
point  of  the  ellipse  (Fig.  80),  then 

that  IS,  m  words  : 

*  To  show  tills  analytically,  let  OF'  =  «e,  and  OZ'  —  ^f ,  and  let  P^{x,  y) 

G 

be  any  point  on  the  ellipse,  as  before.  Equation  (5),  of  Art.  110,  gives  the 
relation  between  x  and  y  ;  expanding  equation  (5),  and  subtracting  4aex 
from  each  member,  it  becomes 

«2g2  _  2  aeic  +  x2  +  ?/2  =  ^2  _  2  aex  +  e^x% 
which  may  be  written 

(ae  -  xy  +  ?/  =  e2  /«  _  x\ 

i.e.,  ¥P^  =  e'^  PL'^  ; 

which  shows  that  P  is  on  an  ellipse  whose  focus  is  i^  and  whose  directrix 

is  D'lDi. 


\  2 


18i 


AjSfALYTIC   GEOMETRY 


[Ch.  VIII. 


If  from  any  'point  on  the  ellipse  a  perpendicular  he  draivn 
to  the  transverse  axis  ;  then  the  square  of  the  distance  from  the 
center  of  the  ellipse  to  the  foot  of  this  perpendicular^  divided  hy 
the  square  of  the  semi-transverse  axis,  plus  the  square  of  the 
perpendicular  divided  by  the  square  of  the  semi-conjugate  axis, 
equals  unity. 

This  geometric  or  physical  property  belongs  to  no  point 
not  on  the  curve,  and  therefore  completely  determines  the 
ellipse.  It  enables  one  to  write  immediately  the  equation  of 
any  ellipse  whose  axes  are  parallel  to  the  coordinate  axes. 

For  example  :  if,  as  in  Fig.  82,  the  major  axis  of  an  ellipse 
is   parallel   to   the  ic-axis,  and   the  center  is  at   the   point 


(7=(A,  ^),  let  P=(x,y)  be  any  point  on  the  curve,  and 
«,  6  be  the  semi-axes,  then 


that  is 


(ag  -  h)^     (y  -7c)2 


1, 


[45] 


which  is  the  equation  of  the  given  ellipse. 

Or  again,  if,  as  in  Fig.  83,  the  major  axis  is  parallel  to 
the  «/-axis  ;  then,  as  before 


CM'    Mr 

ca'     cb' 


1, 


112.] 


THE  CONIC  SECTIONS 


185 


which  is  the  equation  of  the  given 
ellipse. 

Equation  [45]  may  be  considered 
a  second  standard  form  of  the  equa- 
tion of  the  ellipse  ;  by  a  change  of 
coordinates  to  a  set  of  parallel  axes 
through  the  center  C  =  (h^  A;),  as 
the  new  origin,  it  can  be  reduced 
to  the  first  standard  form. 

By  Art.   110  the   distance   from 
the  center  of  an  ellipse  to  its  focus 
is   ae  ;    but    since    h^  =  a^(l  —  e^)* 
[Art.   110,   eq.    (T)],    therefore    ae  =  Va^  —  5^ ;    hence,    in 
Figs.  82  and  83, 


K 


L 
Fig.  83. 


O    X 


Again,  the  equation  of  an  ellipse,  in  either  standard  form, 
gives  the  semi-axes  as  well  as  the  center  of  the  curve,  there- 
fore the  positions  of  the  foci  are  readily  determined  from 
either  standard  form  of  the  equation. 

EXERCISES 

Construct  the  following  ellipses,  and  find  their  equations: 

1.  given  the  focus  at  the  point  (  —  1,  1),  the  equation  of  the  directrix 
a:  —  ?/  +  3  =  0,  and  the  eccentricity  \  (cf .  Art.  109)  ; 

2.  given  the  focus  at  the  origin,  the  equation  of  the  directrix  x  =  —Q, 
and  the  eccentricity  ^ ; 


*  The  student  should  observe  that  h  is  the  semi-minor-axis  and  not  nec- 
essarily the  denominator  of  y'^  in  the  standard  forms  of  the  equation  of  the 
ellipse —  [44],  [45],  or  [46]  ;  he  should  also  observe  that  the  foci  are  always 
on  the  major  axis. 


186  ANALYTIC   GEOMETRY  [Ch.  VIII 

3.  given  the  focus  at  the  point  (0,  1),  the  equation  of  the  directrix 
y  —  25  =  0,  and  the  eccentricity  \ ; 

4.  given  the  center  at  the  origin,  and  the  semi-axes  V2,  VS.     Find 
also  the  latus  rectum. 

Find  the  equation  of  an  ellipse  referred  to  its  center,  whose  axes  are 
the  coordinate  axes,  and 

5.  which  passes  through  the  two  points  (2,  2)  and  (3,  1). 

6.  whose  foci  are  the  points  (3,  0),  (~3,  0),  and  eccentricity  \. 

7.  whose  foci  are  the  points  (0,  6),  (0,  -6),  and  eccentricity  |. 

8.  whose  latus  rectum  is  5,  and  eccentricity  f . 

9.  whose  latus  rectum  is  8,  and  the  major  axis  10. 

10.  whose  major  axis  is  18,  and  which  passes  through  the  point  6,  4. 

Draw  the  following  ellipses,  locate  their  foci,  and  find  their  equations : 

11.  given  the  center  at  the  point  (3,  ~2),  the  semi-axes  4  and  3,  and 
the  major  axis  parallel  to  the  a:-axis  (cf.  Art.  112)  ; 

12.  given  the  center  at  the  point  (~8,  1),  the  semi-axes  2  and  5,  and 
the  major  axis  parallel  to  the  y-axis ; 

13.  given  the  center  at  the  point  (0,  7),  the  origin  at  a  vertex,  and 
(2,  3)  a  point  on  the  curve ; 

14.  given  the  circumscribing  rectangle,  whose  sides  are  the  lines 
a:-fl  =  0,  2a;  —  3  =  0,  ?/-f6  =  0,  3y-f4  =  0;  the  axes  of  the  curve 
being  parallel  to  the  coordinate  axes. 

15.  If  h  becomes  more  and  more  nearly  equal  to  a,  what  curve  does 
the  ellipse  approach  as  a  limit  ? 

113.  Every  equation  of  the  form  Anc^  +  By^  +  ^Goc  +  ^Fy 
-f  C  =  O,  in  which  A  and  B  have  the  same  sign,  represents 
an  ellipse  whose  axes  are  parallel  to  the  coordinate  axes. 
Equations  [44],  [45],  and  [46],  obtained  for  the  ellipse,  are 
all,  when  expanded,  of  the  form 

Ax^  +  By'^^-'iax-{-2Fy  +  Q=0,,     .     .     (1) 

where  A  and  B  have  the  same  sign,  and  neither  of  them  is  zero. 
Conversely,  an  equation  of  this  form  represents  an  ellipse 


112-113.] 


THE  CONIC  SECTIONS 


187 


whose  axes  are  parallel  to  the  coordinate  axes.  As  in 
Art.  107,  a  numerical  case  will  first  be  examined,  and  then 
the  general  equation  taken  up  in  a  similar  manner. 

Example.     Given  the  equation  4:  x^  +  9  y^  ~  IQ  x  -\-  IS  y  —  11  =  0,  to 
show  that  it  represents  an  ellipse,  and  to  find  its  elements.     Completing 


the  square  for  the  terms  in  x,  and  also  for  those  in  y,  and  transposing, 
this  equation  becomes 

4a;2  -  16  a;  +  16  +  9 2/2  +  18  y  +  9  =  11  +  16  +  9, 

that  is,  4  (a:  -  2)2  +  9  (y  +  1)2  =  36 ; 


hence 


(X  -  2)2     (y  +  iy^^ 


32 


22 


This  equation  is  of  the  form  [45],  and,  therefore,  its  locus  has  the 
geometric  property  given  in  Art.  112,  and  is  an  ellipse.  Its  center  is 
the  point  (2,  —1);  its  major  axis  is  parallel  to  the  x-axis,  of  length  6; 
its  minor  axis  is  of  length  4 ;  the  foci  are  the  points 

F'=(2-V5,-l),  F=(2+V5,-l)r 
and  the  equations  of  the  directrices  are,  respectively, 


2  + 


9_ 


x  =  2- 


V5 


Following   the    method  illustrated   above,  of   completing 
the  squares,  the  general  equation  (1)  may  be  written 


188  AJSrALYTIC   GEOMETRY  [Ch.  YIII. 

that  is, 


Aj  V        BJ  AB 

which  becomes,  if  the  second  member  be  represented  by  K, 

"^^  +^     /^=l.     ...      (2) 


K  K 

A  B 

Comparing  this  equation  with  [45]  or  [46],  it  is  seen  to 
express  the  geometric  relation  of  Art.  112,  and  therefore 
represents  an  ellipse.     Its  axes  are  parallel  to  the  coordinate 

— J ,  —^\  and  the  lengths 
of  the  semi-axes  are 

The  foci  and  directrices  may  be  found  as  above. 

Note,  li  A  =  B,  then  equation  (1)  represents  a  circle  (Art.  79).  If 
ABC  >  BG^  +  AF'^,  equation  (1)  having  been  written  with  A  and  B 
positive,  then  no  real  values  of  x  and  y  can  satisfy  equation  (2),  which 
is  only  another  form  of   equation  (1),  and   it   is   said   to  represent  an 

imaginary  ellipse.     If  ABC  =  BG^  +  AF'^,  then  x  = ,   and  y  = 

are  the  only  real  values  that  satisfy  equation  (2)  ;  in  that  case,  this  equa- 
tion is  said  to  represent  a  point  ellipse ;  or,  from  another  point  of  view, 

(C        F\ 
—  — ,  — -  j.    Each 

of  the  above  may  be  regarded  as  a  limiting  form  of  the  ellipse. 

EXERCISES 

Determine,  for  each  of  the  following  ellipses,  the  center,  semi-axes, 
foci,  vertices,  and  latus  rectum ;  then  sketch  each  curve. 


113-114. J  THE  CONIC  SECTIONS  189 

1.  Sx'^  +  d?/  -Qx  -27y  +  2  =  0. 

2.  4:x^ +  f -Sx +  2y +  1  =0. 

3.  x^  +.15 y2  +  4  a;  +  60?/  +  15  =  0. 

4.  By  completing  the  squares  of  the  a:-terms  and  of  the  y-terms,  and  a 
suitable  transformation  of  coordinates,  reduce  the  equations  of  exercises 
1,  2,  and  3  to  the  standard  form  [44]. 

114.    Reduction  of  the  equation  of  an  ellipse  to  a  standard  form. 

It  is  now  evident  that,  if  the  directrix  and  focus  of  an  ellipse  are 
known,  as  in  the  example  of  Art.  109,  the  transformation  of  coordinates 


Fig.  85. 


which  is  necessary  to  reduce  the  equation  to  a  standard  form  can  easily 
be  determined.  To  illustrate ;  the  ellipse  of  eccentricity  f,  with  focus  at 
F=(2,  -I),  and  having  for  directrix  the  line  D'D.  whose  equation  is 
X  -{-2i/  =  6,  has  for  its  equation  (Art.  109) 

41  x^  -lQxy  +  29  y'^  -  140  a;  +  170  y  +  125  =  0. 

Its  axis  FZ,  perpendicular  to  D'D,  has  the  equation  2  x  —  y  =  5,  and 
cuts  the  rr-axis  at  the  angle  tan-^  2.  If  then  the  coordinate  axes  are  rotated 
through  the  angle  tan-^2,  the  equation  will  be  reduced  to  the  second 
standard  form.  Again,  Z  may  be  found  as  the  intersection  of  the 
directrix  and  axis;  it  is  the  point  (3,  1).    Then  A  and  A',  the  vertices 


190  ANALYTIC  GEOMETRY  [Ch.  VIII. 

of  the  ellipse,  divide  FZ  internally  and  externally  in  the  ratio  f ;  hence 
(Art.  30)  these  coordinates  are  (-^/,  -|),  (0,  ~5).  Also  C,  the  center 
of  the  ellipse,  is  the  point  (f,  ~-^#).  If  the  origin  be  next  transformed 
to  the  point  C,  the  equation  will  be  reduced  to  the  first  standard  form. 

12 

Since  the  axis  A  A'  is  of  length  -^,  and  the  eccentricity  is  |,  the  semi- 

6  V5 

axes  are  —  and  2 ;  hence  the  reduced  equation,  with  C  as  origin  and 

\/5 
CA  as  a;-axis,  will  be 

—  +  ^  =  1. 

36  4 

The  problem  of  reducing  to  standard  form  the  equation  of  an  ellipse, 
when  the  directrix  is  not  known,  will  be  postponed  to  Chapter  XII. 


EXERCISES 

Find,  and  reduce  to  the  first  standard  form,  the  equation  of  the  ellipse  : 

1.  with  focus  at  the  point  (1,  ~o),  with  the  line  x  +  y  =  7  for  direc- 
trix, and  eccentricity  ^; 

2.  with  focus  at   the  point    (a,  b),  the   line  -  +  ^  =  1  for  directrix, 

/  a     6 

and  eccentricity  —  (where  Z<w). 
n 


III.    THE   HYPERBOLA 
Special  Equation  of  the  Second  Degree 

115.  The  hyperbola  defined.  An  hyperbola  is  the  locus  of 
a  point  which  moves  so  that  the  ratio  of  its  distance  from  a 
fixed  point,  called  the  focus,  to  its  distance  from  a  fixed  line, 
called  the  directrix,  is  constant  and  greater  than  unity.  The 
constant  ratio  is  the  eccentricity  of  the  hyperbola.  This 
curve  is  the  conic  section  with  eccentricity  e  >  1  (cf . 
Art.  48). 


114-116.] 


THE  CONIC  SECTIONS 


191 


Since  the  hyperbola  differs  from  the  ellipse  only  in  the 
sign  of  1  —  e\  which  is  +  in  the  ellipse  and  —  in  the  hyper- 
bola, the  standard  equation  of  the  hyperbola  can  be  derived 
by  the  method  of  Art.  110  ;  and  it  will  be  found  that  with 
choice  of  axes  and  notation  as  there  given,  the  results  given 
in  eqs.  (1),  (2),  and  (3)  of  that  article  apply  equally  to  the 
hyperbola.  If  now,  since  1  —  e^  is  negative,  the  substitution 
52  —  ^2(^2  _  1)  is  made,  equation  (6)  (p.  181)  will  become 


or 


[47] 


which  is  the  simplest  equation  of  the  hyperbola.     For  variety, 
this  equation  will  be  obtained  by  a  different  method. 

116.  The  first  standard  form  of  the  equation  of  the  hyper- 
bola. Let  F  be  the  focus, 
D'D  the  directrix,  and  e  the 
eccentricity  of  the  curve. 
Take  B'D  as  the  ^/-axis,  with 
the  perpendicular  OFX  upon 
it,  through  the  focus,  as  the 
a;-axis.  Let  2p  denote  the 
given  distance  OF^  and  let 

P  =  (x,y-) 

be  any  point  of  the  locus,  with  coordinates  LP  and  MP. 


Then 
but 


FP  =  eLP 


[geometric  equation] 


FP  =  V(x  -  2^)2  +  2/2,  and  MP  =  x ; 
(x—  2^)2  +  y2  =  A^, 
that  is,  (e2  _  1)2;2  + 2/2  +  4^(92:- 4  jt>2  =  0,    .      .      .      (1) 

which   is   the    equation   of    the    hyperbola    referred   to    its 
directrix  and  principal  axis  as  coordinate  axes  (cf.  Art.  48). 


192  ANALYTIC  GEOMETRY  [Ch.  VIII. 

The  curve  cuts  the  a:-axis  in  two  points,  A  =  (a;^,  0), 
and  A' =  (^x^,  0),  —  the  vertices  of  the  hyperbola,  —  whose 
abscissas  are  determined  by  the  equation 

(^2  -  l)x^  +  4:px  +  4:p^  =  0. 

The  abscissa  of  (7,  the  middle  point  of  the  segment  AA\ 
is,  therefore, 

0C  =  ^1±^  =  ^^      (Art.  11); 

2  e^  —1 

hence  the  center  is  on  the  opposite  side  of  the  directrix  from 
the  focus. 

Now  transform  equation  (1)  to  a  parallel  set  of  axes 
through  C;  the  equations  for  transformation  are  (Art.  71) 

x  =  x' ^  ^^,    and    ?/=?/'; 

substituting   these   values,   and    removing   accents,   eq.    (1) 
becomes 

which  reduces  to    (^e^  —  l')x^  +  ^^  =  ^   y 

that  is,  4^2g2         4^V  ~  *       '       ^  ^ 

(^2  - 1)2     ^2Tri 

If  these  denominators  are  represented  by  c^  and  W'  respec- 
tively, 2.e.,  if 

(.2-1) 

then  P  =  a\e^  - 1),        .  .  .  (4) 

and  equation  (2)  may  be  written  in  the  simple  form 


a^  =  r,,2      i\2^  ^^^  ^^  "^    2      1  '      '      •      '      ('^) 


116-117.]  THE  CONIC  SECTIONS  193 

the  standard  equation  of  the  hyperbola.  Every  equation 
representing  an  hyperbola  can  be  reduced  to  this  form,  as  is 
shown  in  Chapter  XII. 

The  distance  from  the  center  to  the  focus  of  the  hyperbola 

^  =  1  is  easily  found  thus  : 

OF  =00+  OF 


g'2_l    '    -^  ^2-1' 

but,  from  equation  (2), 

a  =   „^   -.^ 

6;^  —  1 

hence  OF  =  ae, 

therefore  the  focus  F  is  the  point  (^ae,  0^.     ,     .     .     (4) 

Similarly  for  the  directrix  : 


00 


2p        a 


—f 


e2  —  1      e 

hence  the  directrix  is  the  line  x =  0.      .     .     .     (5) 

e 

As  above  defined,  h  is  real,  and  its  value  is  known  when  a 
and  e  are  known.     In  Fig.   86, 

OB  =  b,  OB'  =  -b,  and    h  =  aVe^-l. 

2  2 

117.  To  trace  the  hyperbola  ^-^,  =  1.  Equation  [47] 
sliows  that : 

(1)  The  hyperbola  is  symmetrical  with  regard  to  the 
a;-axis;  that  is,  with  respect  to  the  line  through  the  focus 
and  perpendicular  to  the  directrix.  This  line  is  therefore 
called  the  principal  axis  of  the  hyperbola  ; 

(2)  The  hyperbola  is  symmetrical  with  regard  to  the 
?/-axis  also;  i.e.,  with  regard  to  the  line  parallel  to  the  di- 
rectrix and  passing  through  the  mid-point  of  the  segment 
cut  by  the  curve  from  its  principal  axis ; 

TAN.  AN.  GEOM.  —  13 


194 


ANALYTIC  GEOMETRY 


[Ch.  VIII. 


(3)  For  every  value  of  x  from  —  a  to  a,  y  is  imaginary ; 
while  for  every  other  value  of  x,  y  is  real  and  has  two 
values,  equal  numerically  but  opposite  in  sign.  But  for 
every  value  of  y,  x  has  two  real  values,  equal  numerically 
and  opposite  in  sign.  When  x  increases  numerically  from  a 
to  00,  then  y  increases  also  numerically  from  0  to  oo. 

These  facts  show  that  no  part  of  the  hyperbola  lies 
between  the  two  lines  perpendicular  to  its  principal  axis  and 
drawn  through  the  vertices  of  the  curve ;  but  that  it  has 
two  open  infinite  branches,  lying  outside  of  these  two  lines. 
The  form  of  the  hyperbola  is  as  represented  in  Fig.  86. 

The  segment  A^A  of  the  principal  axis,  intercepted  by  the 
curve,  is  called  its  transverse  axis.     The  segment  B^B  of  the 

second  line  of  S3anmetry  (the 
«/-axis),  where  ^'0  =  OB  —  6, 
is  called  the  conjugate  axis ; 
and  although  not  cut  by  the 
hyperbola,  it  bears  impor- 
tant relations  to  the  curve. 
From  the  symmetry  of  the 
hyperbola,  with  respect  to 
these  axes,  it  follows  that  it 
is  also  symmetrical  with  re- 
spect to  their  intersection  0, 
the  center  of  the  curve.  It  follows  also  that  there  is  a  sec- 
ond focus  at  the  point  (—  ae,  0),  and  a  second  directrix  in 

ft 

the  line  a:  -h  -  =  0  on  the  negative  side  of  the  conjugate  axis, 
e 

and  symmetrical  to  the  original  focus  and  directrix.     (See 
Art.  Ill,  foot-note.) 

The  latus  rectum  of  the  hyperbola  is  readily  found  to  be 
2  62 


a 


(cf.  Arts.  105,  111). 


117-118.] 


THE   CONIC   SECTIONS 


195 


118.  Intrinsic  property  of  the  hyperbola.  Second  standard 
equation.  Equation  [47]  states  a  geometric  property  which 
belongs  to  every  point  of  an  hyperbola,  whatever  the  coordi- 
nate axes  chosen,  and  to  no  other  point ;  and  which  therefore 
completely  defines  the  hyperbola.  With  the  figure  and 
notation  of  Art.  117,  equation  [47]  states  (Fig.  87) 


om^mP 


=  1, 


a  property  entirely  analogous  to  that  of  Art.  112  for  the 
ellipse.     It  enables  one  to  write  at  once  the  equation  of  an 


FIG..89. 


hyperbola  with  given  center  and  semi-axes,  and  axes  parallel 
to  the  coordinate  axes. 

For  example,  if  the  transverse  axis  is  parallel  to  the 
a;-axis,  as  in  Fig.  88,  and  the  center  at  the  point  (7=  (A,  A;), 
and  a  P  =  (x,  y)  is  any  point  on  the  curve  ;  then 


CJ?     cP 


1, 


a" 


[48] 


196  ANALYTIC  GEOMETRY  [Ch.  VIII. 

which  is  the  equation  of  the  hyperbola,  with  a  and  h  as  semi- 
axes. 

Again,  if  the  transverse  axis  is  parallel  to  the  «/-axis,  as  in 
Fig.  89,  with  the  center  at  the  point  (7i,  A;),  the  equation  of 
the  hyperbola  will  be  found  to  be 


Note  1.  That  the  expressions  obtained  on  p.  193  for  the  distances 
fi-om  the  center  to  the  focus  and  the  directrix,  of  hyperbola  [47],  are 
equally  true  for  hyperbolas  [48]  and  [49]  follows  from  the  fact  that 
those  expressions  involve  only  a,  h,  and  e ;  moreover,  equation  (4)  of 
Art.  116  determines  e  in  terms  of  a  and  b ;  hence,  for  all  these  hyper- 

bolas,  e^  =  — — — ,  the  distances  from  the  center  to  the  foci  are  given  by 

CF=ae=±  y/a^TTS 
and  those  to  the  directrices  by 

CZ  =~  = 


±  y/a^  +  b^ 


Note  2.  It  should  be  noticed  that  in  equations  [47],  [48],  [49],  the 
negative  term  involves  that  one  of  the  coordinates  which  is  parallel  to 
the  conjugate  axis. 

EXERCISES 

1.  Find  the  equation  of  the  hyperbola  having  its  focus  at  the  point 
(~1,  -1),  for  its  directrix  the  line  Sx  —  y  =  7,  and  eccentricity  f.  Plot 
the  curve  (cf .  Art.  105,  and  Art.  109,  Ex.) . 

Find  the  equation  of  the  hyperbola  whose  center  is  at  the  origin  and 

2.  whose  semi-axes  equal,  respectively,  5  and  3  (cf.  Art.  116,  [47])  ; 

3.  with  transverse  axis  8,  — the  point  (20,  5)  being  on  the  curve; 

4.  the  distance  between  the  foci  5,  and  eccentricity  V^ ; 

5.  with  the  distance  between  the  foci  equal  to  twice  the  transverse 
axis. 

Find  the  equation  of  an  hyperbola 

6.  with  center  at  the  point  (3,  -2),  semi-axes  4  and  3,  and  the  trans- 
verse axis  parallel  to  the  a;-axis.     Plot  the  curve  (cf.  Art.  118) ; 


118-119.]  THE  CONIC  SECTIONS  197 

7.  with  center  at  the  point  (-3,  -4),  semi-axes  6  and  2,  and  the 
transverse  axis  parallel  to  the  y-axis.     Plot  the  curve. 

8.  Find  the  foci  and  latus  rectum  for  the  hyperbolas  of  exercises 
6  and  7. 

9.  By  a  suitable  transformation  of  coordinates,  reduce  the  equations 
of  exercises  6  and  7  to  the  standard  form  —■  —  f-=  1. 

10.   Find  the  foci  of  the  hyperbolas 

^^     25      9   ~    '      ^^^     4       9         '^^^9       4 
Plot  the  curves  (/3)  and  (y). 

119.   Every  equation  of  the  form  Ax^  +  Bij^  +  2Gx  +  2Fy 

+  C  =  0,  in  which  A  and  B  have  unlike  signs,  represents  an 
hyperbola  whose   axes   are    parallel  to  the  coordinate  axes. 

When  cleared  of  fractions  and  expanded,  the  three  equations 
found  for  the  hyperbola  are  of  the  form 

Ax^-{-Bf-{-2ax-{-2F^-\-  0=0,    .     .     .     (1) 

where  A  aiid  B  have  opposite  signs,  and  neither  of  them  is  zero. 
Conversely,  it  will  now  be  shown  that  every  equation  of  this 
form  represents  an  hyperbola,  whose  axes  are  parallel  to  the 
coordinate  axes.  A  numerical  case  will  be  examined  first, 
and  then  the  general  equation. 

Example.  To  show  that  the  equation  9  a;^  -  4  ?/2  -  18  a;  +  24  ?/  -  63  =  0 
represents  an  hyperbola,  and  to  find  its  elements.  Transposing  the  con- 
stant term,  and  completing  the  squares  of  the  x-terms  and  ?/-terms,  the 
equation  may  be  written 

9(a;_l)2_4(3/_3)2^36^ 

Since  this  equation  is  of  the  form  [48],  its  locus  has  the  geometric 
property  given  in  Art.  118,  and  therefore  represents  an  hyperbola.  Its 
center  is  at  the  point  (1,  3),  its  transverse  axis  is  parallel  to  the  a,--axis, 
of  length  4,  and  its  conjugate  axis  is  of  length  6.  The  eccentricity  is 
e  =  l  Vl3,  the  foci  are  at  the  points  (1  -  Vl3,  3)  and  (1  +  vT3,  3)  ;  and 
the  directrices  are  the  lines  whose  equations  are 


198  ANALYTIC  GEOMETRY  [Ch.  VIII. 

Following  the  method  illustrated  in  the  numerical  example, 
the  general  equation  (1)  may  be  written  in  the  form 


K        '        K 
A  B 

wherein  (cf.  Art.  113,  p.  188), 

BG^+AF'^-ABO 


+        ^       =^h      .       .      »     (2) 


K= 


AB 


Since  A  and  B  have  opposite  signs,  the  two  terms  in  the 
first  member  of  this  equation  are  of  opposite  signs  ;  the 
equation  is  therefore  in  the  form  of  [48]  or  [49],  and  repre- 
sents an  hyperbola.     Its  axes  are  parallel  to  the  coordinate 

axes,  its  center  is  the  point  [  — — ,  — -  ],  and  its  semi-axes 
are^±  — *  and  ^±-. 

Note.  Since  A  and  B  have  opposite  signs,  eqnation  (2),  which  is 
only  another  form  of  equation  (1),  always  represents  a  real  locus ;  it  is  an 
hyperbola  proper  except  when  ABC  =  BG^  -\-  AF\  and  it  then  represents 
a  pair  of  intersecting  straight  lines  (cf .  Art.  67). 

It  is  clear  that  the  method  shown  for  the  ellipse  in  Art.  114 
can  be  applied  equally  well  to  the  hyperbola,  to  reduce  any 
equation  of  this  curve  to  the  standard  form,  when  the  direc- 
trix is  known.  The  problem  of  reducing  to  the  standard 
form  the  general  equation  of  an  ellipse,  when  the  directrix 
and  focus  are  not  known,  is  considered  in  full  in  Chapter  XII. 

*  That  sign  ( +  or  — )  which  makes  the  fraction  positive  is  to  be  used. 


119-120.]  THE  CONIC  SECTIONS  199 

EXERCISES 

Determine  for  each  of  the  following  hyperbolas  the  center,  semi-axes, 
foci,  vertices,  and  latus  rectum : 

1.  16  a:2  -  8  2/2  +  64  a;  -  36  2/  +  10  =  0 ; 

2.  x^-5y^  +  157j -10x-\-l-0; 

3.  2x  +  Qij  +  Sy^  =  x^  +  7. 

4.  Reduce  the  equations  of  exercises  1,  2,  3,  to  the  standard  form 

^  =  1.     Sketch  each  curve. 

120.  Summary.  In  the  preceding  articles  it  has  been 
shown  that  the  special  equation  of  the  second  degree, 

Ax^  +  Bf  +  2ax  +  2Fy  +(7=0, 

always  represents  a  conic  section,  whose  axes  are  parallel  to 
the  coordinate  axes.  There  are  three  cases,  corresponding 
to  the  three  species  of  conic. 

(1)  The  parabola  :  either  A  or  B  is  zero.  In  exceptional 
cases  this  curve  degenerates  into  a  pair  of  real  or  imaginary 
parallel  straight  lines,  and  these  may  coincide.       [Art.  107] 

(2)  The  ellipse  :  neither  A  nor  B  is  zero,  and  they  have 
like  signs.  In  exceptional  cases  this  curve  degenerates  into 
a  circle,  a  point,  or  an  imaginary  locus.        [Art.  113,  Note] 

(3)  The  hyperbola  :  neither  A  nor  B  is  zero,  and  they 
have  unlike  signs.  In  exceptional  cases  this  curve  degener- 
ates into  a  pair  of  real  intersecting  lines.  [Art.  119] 

The  ellipse  and  hyperbola  have  centers,  and  therefore  are 
called  central  conies,  while  the  parabola  is  said  to  be  non- 
central  ;  although  it  is  at  times  more  convenient  to  consider 
that  the  latter  curve  has  a  center  at  infinity,  on  the  princi- 
pal axis  (cf.  Appendix,  Note  E). 

The  equation  for  each  conic  has  two  standard  forms,  which 
state  a  characteristic  geometric  property  of  the  curve,  and  to 
which  all  other  equations  representing  that  species  can  be 


200  A^^ALTT1C  GEOMETRY  [Ch.  VIII. 

reduced.  These  standard  forms  are  the  simplest  for  stiidy- 
mg  the  curves  ;  l»ut  the  student  must  discriminate  carefully 
between  general  results  and  those  which  hold  only  when  the 
equation  is  in  the  standard  form. 

IV.     TANGENTS,  NORMALS,  POLARS,  DIAMETERS,  ETC. 

121.  Since  the  equation 

Ax'-]- Bf  +  2ax-{-2F^-h  0=0  .     .     .     (1) 

always  represents  a  conic  whose  axes  are  parallel  to  the 
coordinate  axes,  and  since  by  giving  suitable  values  to  the 
constants  A,  B^  (7,  F^  and  (7,  equation  (1)  may  represent  any 
such  conic,  therefore,  if  the  equations  of  tangents,  normals, 
polars,  etc.,  to  the  locus  of  equation  (1)  can  be  found,  inde- 
pendent of  the  values  that  A,  B^  etc.,  may  have,  these  equa- 
tions will  represent  the  tangents,  etc.,  when  any  special 
values  whatever  are  given  to  the  constants  involved. 
In  the  next  few  articles  such  equations  will  be  found. 

122.  Tangent  to  the  conic 

Ax^  +  J52/2  +  ^Gx^-'^Fy-\-C  =  0 

in  terms  of  the  coordinates  of  the  point  of  contact :  the  secant 
method.  The  definition  of  a  tangent  has  already  been  given 
(Art.  81),  and  the  method  to  be  employed  here  in  finding 
its  equation  is  the  one  which  was  used  in  Art.  84.  That 
article  should  now  be  carefully  re-read. 

Let  the  given  conic,  z.e.,  the  locus  of  the  equation, 

Ax'  +  By''+2ax^-2Fy  +  C=0,    .    .    .    (1) 

be  represented  by  the  curve  BMK\  and  let  P^  =  0^1,  y\)  be 
the  point  of  tangency. 


120-122.] 


THE  CONIC  SECTIONS 


201 


Through  Pi  =  (a:^,  y^)  draw  a 
secant  line  LM,  and  let  Pg^  (^2'  ^2) 
be  its  other  point  of  intersection 
with  the  locus  of  equation  (1).  If 
the  point  P2  moves  along  the  curve 
until  it  comes  into  coincidence  with 
Pi,  the  limiting  position  of  the  se- 
cant LMis  the  tangent  PiT. 

The  equation  of  the  line  LM  is 


Fig. 90. 


y  -  y\=^ 


-^{X  —  Xy). 


(2) 

X2  Xy 

If  now  P2  approaches  Pi  until  x^  =  Xi  and  7/2  =  ?/i,  equa- 
tion (2)  assumes  the  indeterminate  form 


(3) 


This  indeterminateness  arises  because  account  has  not  yet 
been  taken  of  the  path  (or  direction)  by  which  P2  shall 
approach  Pi,  and  it  disappears  immediately  if  the  condition 
that  Pi  and  P2  are  points  on  the  conic  (1)  is  introduced. 
Since  Pi  and  P2  are  on  the  conic  (1), 

therefore      Ax^'  +  Bij^'  +  2  Gx^  ^  2  F^^  +  C  =  0,   .   .   .   (4) 

and  AX2'  +  %2'  +  2ax2-^2Fi/2  +  C=0,   .   .   .   (5) 

Subtracting  equation  (4)  from  equation  (5),  transposing, 
factoring,  and  rearranging  [cf.  Art.  84,  equations  (8),  (9), 
and  (10)],  the  result  may  be  written 

y2  —  yi^     M^i  +  3^2)+  2  (^ 

X2  -  xi  P(yi  +  ^0  +  2  P' 


C^) 


If  this  value  of  ^ ^  is  substituted  in  equation  (2),  the 

result  is  ^        ^ 

^      ^'-      ^(^i  +  y2)  +  2p("      "^^'    "  '  ^'^ 


202       ^  ANALYTIC  GEOMETRY  [Ch.  VIII. 

which  is  the  equation  of  the  secant  line  LM  of  the  given 
conic  (1). 

If  now  this  secant  line  be  revolved  about  Pj  until  P2 
comes  into  coincidence  with  Pj,  i.e.,  until  x.2  =  Xi  and  ^9  =  ^1? 
this  equation  becomes 

which  is,  therefore,  the  equation  of  the  tangent  line  PiT  at 
the  point  Pi.  This  equation  (8)  can  be  put  in  a  much 
simpler  and  more  easily  remembered  form,  thas  : 

Clearing  equation  (8)  of  fractions,  and  simplifying,  it  may 
be  written 

AxiX  +  B^i7/  +  ax-\-F^=Ax^^  +  Bi/^^+ax^+Fi/^,    ...  (9) 

but,  from  equation  (3), 

Ax^'  +  P?/i'  +  (^x,  +  Ft/,=  -  ax,  -  Fy,  -  C. 

hence  substituting  this  value  in  the  second  member  of  equa- 
tion (9)  that  equation  becomes 

Ax^x  +  By^y  ^  Gx^-Fy  =  -  Gx,- Fy,- C,       .     .      .     (10) 

and,  by  transposing  and  combining,  this  may  be  written, 

^a5ia5+ 1^2/12/ +  G(a?+ici)  + 1^(2/ +  1/1)  +  e  =  0.*     .      .      .      [50] 

This  is,  then,  the  equation  of  the  tangent  to  the  conic 

Ax^  -\-  By''  ^  "l  ax  ^  "IFy  -\-  0=  0, 

whatever  the  values  of  the  coefficients  A^  P,  6^,  P,  and  0 
may  be  ;  the  point  (x,^  y,)  being  the  point  of  contact. 

If  J.  =  0,  P=l,  a=  -2p,  F=0  and  (7  =  0,  then  the  equa- 
tion of  this  conic  becomes  y^  =  4:px,  and  the  equation  of  the 
tangent  becomes,  yyi  =  2p(x  + x,);  similarly  for  any  other 
special  form  of  the  equation  of  the  conic. 

*  Compare  note,  Art.  84,  ((3). 


122-123.] 


THE  CONIC  SECTIONS 


203 


123,  Normal  to  the  conic  Ax^  +  By^  +  2Gx  +  2Fy +C=0, 
at  a  given  point.  The  normal  to  a  curve  has  been  de- 
fined (Art.  81)  as  a  straight  line  perpendicular  to  a  tan- 
gent, and  passing  through  the  point  of  contact.  Therefore, 
to  obtain  the  equation  of  a  normal  to  a  conic,  at  a  given 
point  on  the  conic,  it  is  only  necessary  to  write  the  equation 
of  the  tangent  to  the  conic  at  that  point  (by  Art.  122),  and 
then  find  the  equation  of  a  perpendicular  to  the  tangent 
which  passes  through  the  point  of  contact  (cf.  Arts.  53, 
62). 

Example.  To  find  the  equation  of  the  normal  to  the 
ellipse 

18  "^  8  ~ 

at  the  point  (3,  2). 

The  equation  of  the  tangent 
at  the  point  (3,  2)  is 

3^     2^_ 
18  "^  8  ~     ' 


Fig.  91. 


I.e. 


2a;  +  3y  =  12. 
The  perpendicular  line  through  (3,  2)  is 

which  is,  therefore,  the  required  normal. 

Similarly,  to  find  the  normal  to  the  conic  whose  equation 

Ax''-hB?/^-i-2ax  +  2Fi/-hO=0,      .     .     .     (1) 

at  the  point  P^  =  (x^,  y-^  on  the  curve.     The  equation  of  the 
tangent  at  P^  is  (Art.  122) 


Ax^x  +  By^y  +  aQc  +  x^^-V  F(^y  +  y^^+C  =  0    . 


(2) 


204       .  ANALYTIC   GEOMETRY  [Ch.  VIII. 

and  its  slope  is,  therefore,  (Art.  58  (2)) 

Ax^  -{-G- 

Hence  the  required  equation  of  the  corresponding  normal 
at  Pi  is  (Arts.  53,  62) 

EXERCISES 

1.  Is  the  line  3  a;  +  2  y  =  17  tangent  to  the  ellipse  16  x"^  +  25  ^2  =  400  ? 

2.  Find  the  equation  of  a  tangent  to  the  conic  x'^  +  b  y'^  —  ^  x  -{-  \0  y 
-4=0,  parallel  to  the  hne  y  =  ^x  +  1  (cf.  Art.  82). 

Write  the  equations  of  the  tangent  and  normal  to  each  of  the  follow- 
ing conies,  through  a  point  {x^,  y^  on  the  curve  (cf.  Art.  122  [50]). 

3.  -2  +  h  =  1- 

5.  x2  =  4p  (?/  -  5)  ;  sketch  the  figure. 

6.  3  x-2  —  5  z/2  +  24  .r  =  0 ;  sketch  the  figure. 

7.  ;r2  4-5  2/2-3a:  +  10^-4  =  0;  sketch  the  figure. 

8.  Derive,  by  the  secant  method  (cf .  Art.  122),  the  tangent  to  the 
parabola  y'^  -  4j!)x;  the  point  of  contact  being  {x^,  y^. 

9.  Derive,  by  the  secant  method,  the  tangent  to  the  ellipse  a;^  +  4  3/^ 
-  8x  +  20?/  =  0;  the  point  of  contact  being  (x^  y^). 

Write  the  equations  of  the  tangents  and  normals  to  each  of  the  fol- 
lowing conies,  at  the  given  point;  also  sketch  each  figure  : 

10.  9  a:2  +  5y2  +  36  ;^  +  20  ?/  +  11  =  0,  at  the  point  (-2,  1)  ; 

11.  9  a;2  +  4  ?/2  +  6  X  +  4  ?/  =  0,  at  the  point  (0,  0)  ; 

12.  /  -  6  ?/  -  8a:  =  31,  at  the  point  (  -3,  -  1); 


*  Since  the  equation  of  the  normal  [51]  is  so  readily  deduced,  in  every 
particular  case,  from  that  of  the  tangent,  and  since  the  latter  is  so  easily 
remembered,  it  is  not  recommended  that  equation  [51]  be  memorized. 


123-124.]  THE  CONIC   SECTIONS  205 

/-pit  flit 

13.  "4+^=1,  at  the  point  (1,  V3)  ; 

14.  3  a:2  +  4  y"^  =  16,  at  the  point  (2,  "1). 

124.  Equation  of  a  tangent,  and  of  a  normal,  that  pass  through  a 
given  point  which  is  not  on  the  conic. 

The  method  to  be  followed  in  finding  the  equation  of  a  tangent,  or  of 
a  normal,  that  passes  through  a  given  point  which  is  not  on  the  conic, 
may  be  illustrated  by  the  following  example ;  the  same  method  is  appli- 
cable to  any  conic  whatever. 

Let  it  be  required  to  find  the  equation  of  that  tangent  to  the  parabola 

y2_6y  _8a,_3i  ^0,         .  .  .  (1) 

which  passes  through  the  point  (~4, -1).  This  point  not  being  on  the 
parabola,  the  method  of  Art.  116  does  not  apply ;  but,  assuming  for  the 
moment  that  it  is  possible  to  draw  such  a  tangent,  let  {x^,  y-^  be  its  point 
of  contact.     The  equation  of  this  tangent  is  (Art.  122) 

yi2/-3(y  +  2/i)-4(:r  +  xO-31=0.        ...       (2) 

Since  this  tangent  passes  through  the  point  (~4,  —1),  therefore  equa- 
tion (2)  is  satisfied  by  the  coordinates  "~4  and  "1, 

i-e.,  -?/, -3(-l  +  ?/i)  -4(-4  +  x0  -31  =0,    .     .     .     (3) 

which  reduces  to  .r^  +  ?/i  +  3  =  0.  .  .  .  (1) 

Equation  (4)  furnishes  one  relation  between  the  two  unknown  con- 
stants x^  and  2/j ;  another  equation  between  these  two  unknowns  is  fur- 
nished by  the  fact  that  (a-j,  y^  is  a  point  on   the  parabola  (1)  ;   this 

equation  is 

y,2  _  62/1 -8a'i- 31  =  0.         .         .         .         (5) 

Solving  between  equations  (4)  and  (5)  gives 

x^  =  -  2±  2  V2    and   y^  =  -lT  2\/2  ; 

hence,  there  are  two  points  on  the  given  parabola  the  tangents  at  which 
pass  through  the  point  ("4,  -1);  their  coordinates  are  (— 2  +  2  a/2, 
-  1  -  2V2)  and  (-2-2  \/2,  -  1  +  2V2);  and  substituting  either  pair 
of  these  values  for  x^  and  ^^  in  equation  (2)  gives  the  equation  of  a 
straight  line  that  is  tangent  to  the  parabola  (1),  and  that  passes  through 
the  point  (~4,  —1). 

So,  too,  if  it  is  desired  to  find  the  equation  of  a  normal  through  a 
point  not  on  the  curve,  it  is  only  necessary  to  assume  temporarily  the  coor- 
dinates of  the  point  on  the  curve  through  which  this  normal  passes,  and 


206 


ANALYTIC   GEOMETRY 


[Ch.  VIII. 


then  Jind  these  coordinates  by  solving  two  equations,  corresponding  to 
equations  (4)  and  (5)  above. 

The  problem  of  finding  the  above  tangent  could  also  have  been  solved 
by  writing  the  equation  of  a  line  through  the  point  (-4,  -1)  (Art.  53) 
and  having  the  undetermined  slope  ?n,  and  then  so  determining  jn  that 
the  two  ]3oints  in  which  this  line  meets  the  parabola  should  be  coincident. 

125.  Through  a  given  external  point  two  tangents  to  a  conic 
can  be  drawn.  This  theorem  can  be  j^roved  in  precisely  the 
same  way  as  tlie  corresponding  theorem  in  the  case  of 
the  circle  (Art.  89)  was  proved.  It  may  also  be  proved  by 
the  method  already  applied  to  the  parabola  in  the  preceding 
article.  Let  the  latter  method  be  adopted.  Suppose  the 
equation  of  the  conic  to  be 

Ax'  +  Bi/'  -{-  2  ax  -}-  2  Ft/  -h  O  =  0;    .    .    .    (1) 

let  the  locus  of  this  equation  be  represented  by  the  curve 
LPiP2L\  and  let  Q=(h^  ¥)  be  the  given  external  point. 

If  Pi  =  (xi,  y{)  is  a  point 
on  IjPiP^U -:  then  the  equa- 
tion of  the  tangent  at  P^  is 

Ax^x  +  Byiy  ^-G-{x^x-^ 

+  Fiy  +  y,)+C=0,     (2) 

and    this   tangent    will   pass 
through  the  point  Q  if 

Ahxi  +  Bky^  -i-  G-(Ji-hXi) 

j^F(k  +  y,)+C=0,    (3) 

But  Pi  being  on  the  locus  of  equation  (1),  its  coordinates 
Xi  and  yi  also  satisfy  equation  (1) ; 

i.e.,  Axi'  +  %i'  +  2axi-{-2Fyi-\-O=0.    .   .   .   (4) 

If  now  equations  (3)  and  (4)  are  solved  for  Xj  and  j/i,  two 
values  of  each  are  found ;  these  values  are  both  imaginary 
if   Q  is  within  the  conic,  they  are  real  but  coincident  if  Q  is 


Fig.  93. 


124-126.] 


THE   CONIC   SECTIONS 


20T 


on  the  conic,  and  they  are  real  and  distinct  if  Q  i.i  outside  of 
the  conic.  This  proves  not  only  the  above  proposition  but 
also  the  fact  that  no  real  tangent  can  be  draAvn  to  a  conic 
through  an  internal  point,  and  that  only  one  tangent  can  be 
drawn  to  a  conic  through  a  given  point  on  the  curve. 

126.  Equation  of  a  chord  of  contact.  If  the  two  tangents 
are  drawn  from  an  external  point  to  a  conic  section,  the 
straight  line  through  the  corre- 
sponding points  of  tangency  is 
called  the  chord  of  contact  cor- 
responding to  the  point  from 
which  the  tangents  are  drawn 
(cf.  Art.  90). 

Let  P^  =  (x^^  y^  be  the  ex- 
ternal point  from  which  the 
two  tangents  are  drawn  ;    T>^= 

Q^^yd  ^^^  ^3  =  (^3'  ^3)'  ti^^ 

points  of  tangency  of  these  tangents  to  the  conic  whose 
equation  is 

Ax^-\-By'^-\-2ax-{-2Fy-\-C  =  0',  ..  .  .  (1) 
it  is  required  to  find  the  equation  of  the  line  through  T^ 
and  T^. 

The  equation  of  the  tangent  at  T^  (cf.  Art.  122)  is 

Ax^x-{-By^y  +  a(x+x^)+F{y-\-y^^^C  =  (),.  .  ,  (2) 
and  the  equation  of  the  tangent  at  T3  is 

Ax^x-^By^y-\-a(^x-\-x^^  +  FQy  +  y^)+C=0.  .  .  .  (3) 
Since  each  of  these  tangents,  by  hypothesis,  passes  through 
Pj,  therefore  the  coordinates  x-^  and  y^  satisfy  both  equation 
(2)  and  equation  (3)  ;  ^.e., 

Ax^x^  +  By^y^  -f-  a(x^  +x,^  + F {y^  + y^^^  C  ==0,.  .  .  (4) 
and  Ax^x^  -f  By^^y^  +  Qi^x^  +  H^^F  (y^  +  y^)^O=0,     (5) 


Y 

£^ 

=^ 

=.k^ 

^ 

0 

m/\ 

X 

Fig 

.93. 

208  ANALYTIC   GEOMETRY  [Ch.  VITI. 

Equations  (4)  and  (5),  respectively,  assert  tliat  the  points 

^2  =  (^2'  1/2)    ^^^    ^3  =  (^3'  ^3) 
are  each  on  the  locus  of  the  equation 

A3Cioc  +  Bijiy  +  G(oc  +  oci)+F(y  +  yi)  +  C  =  0,      .      .      [52] 

But  equation  [52]  is  of  the  first  degree  in  the  two  vari- 
ables X  and  ^,  hence  (Art.  57)  its  locus  is  a  straight  line ; 
i.e.,  [52]  is  the  equation  of  the  straight  line  through  T^  and 
Tg,  which  was  to  be  found. 

Note  1.  The  equation  [52]  of  the  chord  of  contact  corresponding  to 
a  given  external  point  (a^j,  ?/j),  and  the  equation  [50]  of  the  tangent 
whose  point  of  contact  is  (x^,  y^)  are  identical  in  form.  This  might  have 
been  expected  because  the  tangent  is  only  a  special  case  of  the  chord  of 
contact,  since  the  chord  of  contact,  for  a  given  point,  approaches  more 
and  more  nearly  to  coincidence  with  a  tangent  when  the  point  is  taken 
more  and  more  nearly  on  the  curve. 

Note  2.  The  present  article  furnishes  another  method  of  treatment 
for  the  question  of  Art.  124.  To  get  the  equations  of  the  tw^o  tangents 
that  can  be  drawn  through  a  given  external  point  to  a  given  conic,  it  is 
only  necessary  to  write  the  equation  of  the  chord  of  contact  correspond- 
ing to  this  point ;  then  find  the  points  in  w^hich  this  chord  of  contact 
intersects  the  conic.  These  are  the  points  of  contact  of  the  required 
tangents,  whose  equation  may  then  be  written  down. 

EXERCISES 

1.  By  first  finding  the  chord  of  contact  (Art.  126)  of  the  tangents 
drawn  from  the  point  (~|,  J/)  to  the  conic 

4x2  +  2/2  +  24:3;  -2y  +  17  =  0, 
find  the  points  of  contact,  and  then  write  the  equations  of  the  tangents 
to  the  conic  at  these  points ;  verify  that  these  two  tangents  intersect  in 
the  point  (-|,  V-)- 

2.  Solve  Ex.  1  by  the  method  of  Art.  124. 

3.  Solve  Ex.  1  by  the  method  of  Art.  89. 

4.  Find  the  equation  of   a  normal  through  the  point  (7,  5)  to  the 

conic 

4x2  4- 3/2  +  24x  -  2  2/ +  17  =  0. 


126-127.] 


THE   CONIC   SECTIONS 


209 


Is  it  possible  to  draw  more  than  one  normal  through  (7,  5)  to  the  given 
conic  ? 

5.  By  the  methods  of  Exs.  1,  2,  and  3,  find  the  equations  of  the 
tangents  through  the  origin  to  the  conic 

3  a;2  -  2  ?/^  =  6  a;  +  8  y  +  6. 

6.  By  the  methods  of   Exs.  1,  2,  and  3,  find  the  equations  of  the 
tangents  through  the  point  ("1,  1)  to  the  conic 

9a;2  +  5y2  +  30a;  +  20y  +  11  =  0. 

7.  Sketch  the  conies  whose  equations  are  given  in  Ex.  1,  5,  and  6. 

8.  Find  the  equations  of  the  tangents  to  the  conic,  x^  +  4  y^  =  4, 
from  the  point  (3,  2). 

9.  Find  the  normal  to   the   conic  x^  +  4  ?/2  =  4,  through  the  point 
(3,  2). 

10.  Solve  ExSc  8  and  9,  by  assuming  the  slope  m  of  the  required 
line  (Art.  53),  and  then  determining  m  so  that  the  two  points  in  which 
the  line  meets  the  given  curve  shall  be  coincident. 

127.  Poles  and  polars.  If  through  any  given  point 
P^=(x-^^^  ?/j),  outside,  inside,  or  on  a  given  conic,  a  secant 
is  drawn,  meeting  the  conic  in  two  points  Q  and  i?,  and 
if  tangents  at  Q  and  H  are  drawn,  they  will  intersect  in 
some  point,  as  P'  =  (a;',  ^').  The  locus  of  P'  as  the  secant 
revolves  about  P^  is  the  polar  of  the  point  P-^  (cf.  Art.  91) 
with  regard  to  the  given  conic  ;  and  P^  is  the  pole  of  that 
locus. 

To  find  the  equation  of  the 
polar  of  a  given  point 

P-^  =Qx^,  «/j), 

with  regard  to  a  given  conic 
whose  equation  is 

+  6^=0,     .     .     .     (1) 

let   QP^R  he    any  position  of 
the   secant    through    P^,     and 


Fig.  91. 


TAN.    AX.    GEOM. 


14 


210 


ANALYTIC  GEOMETRY 


[Ch.  VIII. 


let  the  tangents  at  Q  and  R  intersect  in  P'  ~  (x\  y').  Then 
the  equation  of  QP^R  (Art.  126)  is 

Ax'x  +  By'y  +  a(x  +  x')  +  F{y  ^y')+O  =  0 (2) 

Since  this  line  passes  through  P^,  therefore  the  coordinates 
Xj^  and  j/j  satisfy  equation  (2), 

i.e.,     Ax^x'  +  Byy+a(x^-\-x')-{-F(y^-hy')+C  =  0,.  ..(3) 

and  equation  (-3)  asserts  that  the  variable  point  P'  =  (x\  y'^ 
lies  on  the  locns  of  the  equation 

Ax^x+By^y+a(x-\-x;)-\-F(iy^-y{)-{-0  =  Q.  .  .  .  (4) 

Equation  (4)  is  of  the  first  degree  in  the  variables  x  and  ?/, 
hence  (Art.  57),  its  locus  is  a  straight  line ;  the  polar  of  P^, 
with  regard  to  the  conic  (1),  i.e.^  the  locus  of  P' ^  is  then 
the  straight  line  whose  equation  is 

Ax^oc  +  Bijiu  +  G  {oc+oc{)  +  F{y  ^y{)  +  C  =  0, .  ,  ,  [53] 

Note.  That  the  equation  of  a  tangent  [50]  and  of  a  chord  of  con- 
tact [52]  have  the  same  form  as  equation  [53]  is  due  to  the  fact  that  a 
tangent,  and  a  chord  of  contact,  are  but  special  cases  of  a  polar. 

128.  Fundamental  theorem.  An  important  theorem  con- 
cerning poles  and  polars  is  :  If  the  polar  of  the  point  P^,  with 


.    Fig.  95. 


127-129.]  THE   CONIC   SECTIONS  211 

regard  to  a  given  conic,  passes  through  the  point  P^,  then  the 
polar  of  P^  with  regard  to  the  same  conic  passes  through  P^. 
Let  the  equation  of  the  given  conic  be 

Ax^  +  By'^  +  2ax  +  '2Fy  +  C=Q,    .     .     .    (1) 

and  let  the  two  given  points  be 

Pi  =  i^v  y\)  aiiti  P2  -  (^2^  ^2)- 

Then  the  equation  of   the  polar  of  P^  with  regard  to  the 
conic  (1)  is  (Art.  127) 

Ax^x^By^y^a(x-Vx^)^P{y-Vy^^C==^\   •  •  •  (2) 

if  this  line  passes  through  P^,  then 

Ax^x^-^By^y^^aix^^x^^Piy^-^y^-^C^^.   .   .  (3) 

But  the  polar  of  P^  with  regard  to  the  conic  (1)  is 

Ax^x-vBy^y^a(x^x^-VP{y-^y^)^C=^,  •  •  •  (4) 

and  equation  (3)  shows  that  the  locus  of  equation  (4)  passes 
through  the  point  P^,  which  proves  the  proposition. 

129.  Diameter  of  a  conic  section.  The  locus  of  the  middle 
points  of  any  system  of  parallel  chords  of  a  given  conic  is 
called  a  diameter  of  that  conic,  and  the  chords  which  that 
diameter  bisects  are  called  the  chords  of  that  diameter. 

For  a  given  conic,  it  is  required  to  find  the  equation  of 
the  diameter  bisecting  a  system  of  chords  whose  slope  is  m. 
Let  the  equation  of  the  given  conic  (B.JK,  Fig.  96)  be 

Ax^  +  By'^^-2ax-\-2Fy  +  C=0,  .     .      .      (1) 

let   the  equation  of  any  one  of  the  parallel  chords  of  slope 
m,  LM  iov  example,  be 

y=7nx-\-h, (2) 

and  let  the  two  points  in  which  it  meets  the  given  conic  be 
P\  =  (^r  Vi)  and  Pi  =  (^2'  y^)' 


212 


ANALYTIC   GEOMETRY 


[Ch.  VIII. 


Then  (Art.  122,  eq.  (6)), 


(3) 


Fig.  96. 


If  ^  ^  (A,  A;)  be  tlie  mid- 
dle point  of  the  chord 
P1P2,  then 

=  ^l±^andyfc=^^; 

substituting  these  values  of 
x-^  H-  x^  and  ^^  +  ^2  ^^^  equa- 
tion (3),  then  clearing  of 
fractions  and  transposing, 
that  equation  becomes 

Ah-^mBk  +  a-^mF  =  ^.     ...      (4) 

But  equation  (4)  asserts  that  the  coordinates  (h,  k}  of 
the  middle  point  of  any  one  of  this  system  of  parallel  chords 
satisfy  the  equation 

Ax  +  mBi/  +  a  +  niF  =  0,  .  .  .  [54] 
which  is  therefore  the  equation  of  the  diameter  whose  chords 
have  the  slope  m. 

EXERCISES 

1.  Find  the  polar  of  the  point  (2,  1)  with  regard  to  the  hyperbola 
^2  _  2  (?/2  +  a;)  —  4  =  0.  Show  that  this  polar  passes  through  (12,  3), 
and  then  verify  Art.  128,  for  this  particular  case,  by  showing  that  the 
polar  of  (12,  3),  with  regard  to  the  given  hyperbola,  passes  through  (2, 1). 

2.  AYrite  the  equation  of  the  chord  of  contact  of  the  tangents  drawn 
through  (2,  1)  to  the  hyperbola  x^  -  2  y'^  -  2  x  -  i  =  0,  then  find  the 
points  in  which  it  meets  the  curve,  get  the  equations  of  the  tangents  at 
these  points,  and  verify  that  they  pass  through  the  given  point  (2,  1). 

3.  By  specializing  the  coefficients  in  equation  [54],  prove  that  the 
diameter  of  a  circle  is  perpendicular  to  the  chords  of  that  diameter. 


129-130.]  THE  CONIC  SECTIONS  213 

Solution.  If  equation  (1)  of  Art.  123  represents  a  circle,  then 
A  =  B,  and  then  equation  [54]  becomes 

1         G  +  mF 

y  =  —  X -. , 

^  111  Am 

i.e.,  the  slope  of  the  diameter  is ;  but  the  slope  of  the  given  system 

m 
of  chords  is  m,  hence  the  diameter  is  perpendicular  to  its  chords. 

4.  By  means  of  eq.  [54],  i.e.,  by  specializing  its  coefficients,  prove 
that  the  diameter  of  a  circle  passes  through  the  center  of  the  circle. 

5.  By  means  of  equation  [54]  prove  that  any  diameter  of  the  ellipse 
3  a:2  +  ?/'2  —  6  x  +  2  ?/  =  0  passes  through  the  center  of  the  ellipse.  Does 
this  property  belong  to  all  ellipses  ?     To  all  conies  ? 

6.  Find  the  equation  of  that  diameter  of  the  hyiDerbola 

a;2  -  4  ?/2  +  16  ?/  +  6  a:  -  15  =  0, 
whose  chords  are  parallel  to  the  line  ?/  =  2  a;  +  10.     Does  this  diameter 
pass  through  the  center  of  the  curve  ? 

7.  Find  the  angle  between  the  diameter  and  its  chords  in  exercise  6. 

8.  Show  that  every  diameter  of  the  parabola  3  2/^  _  1(3  x  +  12  ?/  =  4 
is  parallel  to  its  axis.     Is  this  a  property  belonging  to  all  parabolas  ? 

9.  Derive,  by  the  method  of  Art.  129,  the  equation  of  that  diameter 
of  the  hyperbola  x^  —  ^y^  +  IQy  +  Qx  —  Id  =  0,  which  bisects  chords 
parallel  to  the  line  3  x  —  4  ?/  =  12. 

130.  Equation  of  a  conic  that  passes  through  the  intersec- 
tions of  two  given  conies.     Let  the  given  conies  be 

S^  =  A^x'  +  B^i/^-2a^x-{-2F^y+C^  =  0,  .   .   .    (1) 

and       S,^  =  A^x^^B,y^'ia^x  +  2F^y+C^_^  =  0',  ...    (2) 

then,  if  k  be  any  constant  whatever, 

S^  +  kS^  =  ^       .           .  .         (3) 

represents  a  conic  whose  axes  are  parallel  to  the  coordinate 
axes  (Art.  120),  and  which  passes  through  the  points  in 
which  the  conies  aS'j  =  0  and  aS'2  =  0  intersect  each  other 
(Art.  41);  i.e.^  aS'^  +  Ar/S'g  =  0  represents  ?i  family  of  conies, 
each  member  of  which  passes  through  the  intersections  of 
aS'j  =  0  and  aS'^,  =  0.     The  parameter  k  may  be  so  chosen  that 


214 


ANALYTIC  GEOMETRY 


[Ch.  VIII. 


the  conic  (3)  shall,  in  addition  to  passing  through  the  four 
points  in  which  8-^  =  0  and  S^  =  ()  intersect,  satisfy  one  other 
condition ;  e.g.^  that  it  shall  pass  through  a  given  fifth  point. 
Moreover,  if  ^^  =  0  and  /S'g  =  0  are  both  circles,  then 
aS'j  +  kS^  =  0  is  also  a  circle  (cf.  Arts.  95  and  96). 


V.     POLAK   EQUATION   OF   THE    CONIC   SECTIONS 

131.  Polar  equation  of  the  conic.  Based  upon  the  "  focus 
and  directrix  "  definition  already  given  in  Art.  48,  the  polar 
equation  of  a  conic  section  is  easily  derived. 

Let  D' D  (Fig.  97)  be  the  given  line  (the  directrix)  and  0 
the  given  point  (the  focus);  draw  ZOR  through  0  and  per- 
pendicular to  2>'i>,  and  let  0  be  chosen 
as  the  pole  and  OR  as  the  initial  line. 
Also  let  P=  (jO,  ^)  be  any  point  on  the 
locus,  and  let  e  be  the  eccentricity. 
Draw  MP  and  OK  parallel,  and  LP 
and  fflT  perpendicular,  to  Z>'-Z>,  and  let 
OK=l\  then 


Fig.  97. 


OP  =  e  •  LP,    [definition  of  the  curve] 
=  eCZO+  OM); 

p  =  el--{-p  cos  0 


This  equation,  when  solved  for  ^,  may  be  written  in  the 
form 

P  =  - 1 ,        .  .  .  [55] 

which  is  the  polar  equation  of  a  conic  section  referred  to 
its  focus  and  principal  axis  ;  e  being  the  eccentricity  and  I 
the  semi-latus-rectum.  If  e  =  1,  equation  {^55^  represents  a 
parabola ;  if  e  <  1,  an  ellipse  ;   and  if  e  >  1,  an  hyperbola. 


130-132.] 


THE  CONIC  SECTIONS 


215 


IN'oTE.  Equation  [55]  shows  that  if  e<l,  i.e.,  if  the  equation  repre- 
sents an  ellipse,  there  is  no  value  of  9  for  which  p  becomes  infinite. 
Therefore  there  is  no  direction  in  which  a  line  may  be  drawn  to  meet  an 
ellipse  at  infinity.  If  e  =  1,  i.e.,  if  the  equation  represents  a  parabola, 
there  is  one  value  of  6,  viz.,  ^  =  0,  for  which  p  becomes  infinite.  There- 
fore there  is  one  direction  in  which  a  line  maybe  drawn  to  meet  a  parab- 
ola at  infinity.  If  e  >  1,  i.e.,  if  the  equation  represents  an  hyperbola, 
there  are  two  values  of  0,  viz.,  0  =  ±  cos~'^  (1  :e),  for  which  p  becomes 
infinite.  Therefore  there  are  tivo  directions  in  which  a  line  may  be  drawn 
to  meet  an  hyperbola  at  infinity. 

The  three  species  of  conic  sections  may  therefore  be  distinguished 
from  each  other  by  the  number  of  directions  in  which  lines  may  be  drawn 
through  the  focus  to  meet  the  curve  at  infinity.  Or,  since  parallel  lines 
meet  at  infinity,  any  point  of  the  plane  may  be  used  instead  of  the  focus. 


6 


COS" 


132.    From  the  polar  equation  of  a  conic  to  trace  the  curve.     Suppose 
e  >  1,  i.e.,  suppose  equation  [55]  represents  an  hyperbola.     When  ^  =  0, 

p  = ,  hence  p  is  negative ;  as  0  increases,  cos  0  decreases,  and  e  cos  0 

1  —  e 

becomes  numerically  more  and  more  nearly  equal  to  1 ;  therefore  p  re- 
mains  negative    and    be- 
comes larger  and  larger; 
p  =  —  CO    when 

1  —  e  cos  ^  =  0, 

i.e.,  when 

say ;       as      0      increases 

through     this     value,     p 

becomes    +  co    and    then 

decreases,     but     remains 

positive,      and      becomes 

equal  to  I  when  0  =  90°;  as  0  increases  through  90°  to  180°,  p  remains 

positive,    but   continues    to  decrease,   reaching  its    smallest    value,  viz. 

p  =  ,  when   0  =  180°;   as  0  increases  from  180°  to  270°,  p  remains 

.  ?-  +  ^          .                                 I  . 

positive  and  increases  from  to   I;    as    0  increases   from   270°   to 

1  +  e 

360°  —  a,  p  increases  from  Z  to  +  go  ;  as  ^  increases  through  360°  —  a,  p 

becomes   —  co  ;  and  finally,  as  0  increases  from  360°  —  a  to  360°,  p  re- 
mains  negative,  but    decreases   numerically,    reaching   the  value   • 

again  when  0  becomes  360°.  ~ 


Fig.  98. 


216  ANALYTIC  GEOMETRY  [Ch.  VIIL 

These  deductions  from  equation  [55]  show  that  the  hyperbola  has 
the  form  represented  in  Fig.  98,  and  that,  as  0  increases  from  0  to  a,  the 
lower  half  .4'TFof  the  infinite  branch  at  the  left  is  traced ;  as  0  increases 
from  a  to  360°  —  a,  the  right  hand  branch  VA  U  is  traced ;  and  as  6  in- 
creases from  360°  —  a  to  360°,  the  upper  half  ^.4'  of  the  left  hand  branch 

is  traced. 

If  0  increases  beyond  360°,  the  tracing  point  moves  along  the  same 
curve ;  this  is  also  true  if  0  changes  from  0°  to  -  360°. 

Note.  To  show  the  identity  of  the  curve  as  traced  in  the  present 
article  and  in  Art.  117,  it  need  only  be  recalled  that 

e  = ,  and  that  I  =  ■ — 

a  a 

These  values  substituted  above  show  that 
a  ^  cos-i(        ^    -]  =  tan-i  (-),  that  OA' =  -  (a  +  VaH^'),  etc. 

EXERCISES 

1.  From  equation  [55] ,  trace  the  parabola. 

2.  From  equation  [55],  trace  the  ellipse. 

3.  By  means  of  equation  [55],  prove  that  the  length  of  a  chord 
through  the  focus  of  a  parabola,  and  making  an  angle  of  30°  with  the 
axis  of  the  curve,  is  four  times  the  length  of  the  latus-rectum. 

4.  By  transforming  from  rectangular  to  polar  coordinates,  derive  the 
polar  equations  of  the  conic  sections  from  their  rectangular  equations. 

EXAMPLES    ON    CHAPTER   VIM 

1.  Find  the  equations  of  those  tangents  to  the  conic  7  x^  -12y^  =  112, 
which  pass  through  the  point  ("9,  7). 

2.  What  is  the  polar  of  the  point  (7,  2)  with  reference  to  the  conic 
16?/2  +  9x2  =  144?  Find  the  equation  of  the  line  which  is  tangent  to 
the  conic  and  parallel  to  this  polar. 

3.  Find    the    polars    of    the    foci    of    the    ellipse  ^+^  =  1,    with 

regard  to  this  ellipse.     Also  for  the  parabola  y'^  =  4:px. 

4.  What  is  the  equation  of  the  polar  of  the  center  of  the  conic 
Ax^  +  S?/2  +  2  Gx  +  2  F?/  +  C  =  0,  with  reference  to  the  conic  ? 

5.  What  is  the  pole  of  the  directrix  of  the  hyperbola  a;^  — 4^/^  =  16, 
with  reference  to  that  curve  ? 


132.]  THE  CONIC  SECTIONS  217 

6.  The  line  y  =  m  (x  —  ae)  passes  througli  the  focus  of  the  central 

conic   —  i:  ^  =  1.     On  what  line  does  its  pole  lie?     Find  the  line  ioin- 

ing  its  pole  to  the  focns.     What  relation  exists  between  this  line  and 
the  given  focal  chord? 

7.  What  is  the  polar  of  the  vertex  of  the  conic 

Ax"^  +  Bi/  +  2  Gx  +  2  Fij  +  C  =  Q, 
■with  reference  to  the  curve  ? 

8.  What  is  the  equation  of  each  common  chord  of  the  two  conies 

16^2+ 9?/2  =  lM,         16a:2-9?/2=  IM? 

Hint.  Use  Art.  130,  equation  3 ;  find  k  so  that  S^  +  kS^  can  be 
factored. 

9.  Prove  that  the   perpendicular   dropped   from  any  point  of   the 
directrix,  to  the  polar  of  that  point,  passes  through  the  focus 

(a)    iovy^  =  ^px.       (/?)    for^±f^=l. 

Using  the  simplest  standard  equations  of  the  conies,  find  for  each 

10.  the  polar  of  the  focus ; 

11.  the  pole  of  the  directrix ; 

12.  the  ratio  of  the  angle  subtended  by  a  chord  at  its  pole,  and  the 
angle  subtended  by  the  same  chord  at  the  focus. 

13.  Find  a  conic  through  the  intersections  of  the  ellipse  4a:2-|-  y'^=  16 
and  the  parabola  ?/2=4a:  +  4,  and  also  passing  through  the  point  2,.  2. 
What  kind  of  a  conic  is  it  ? 

14.  Show  that  the  curves h  —  =  1  and ^  =  1  have  the  same 

16      7  4       5 

foci,  and  that  they  cut  each  other  at  right  angles. 

15.  Find  the  vertices  of  an  equilateral  triangle  circumscribed  about 
the  ellipse  9  x^  +  16  ^/^  =  144,  one  side  being  parallel  to  the  major  axis 
of  the  curve. 

16.  Find  the  normal  to  the  conic  3  x^  +  ?/2  —  2  x  —  ?/  =  1,  making  the 
angle  tan-^  (f )  with  the  x-axis. 

17.  Show  that  the  locus  of  the  pole,  with  respect  to  the  parabola  ?/^  =  4  ax, 
of  a  tangent  to  the  hyperbola  x'^  —  y'^  —  a%  is  the  ellipse  4  x^  +  ?/2  =  4  a^, 

18.  Show  that  — -] ^ =  1,    where  k  is  an   arbitrarv  con- 

a^  —  ^•2      h^  —  k'^ 

stant,  represents  an  ellipse  having  the  same  foci  as  ^  +  -^  =  1    when 


218  AliALYTIC  GEOMETRY  [Ch.  VIII.  132. 

k^<b^;    but   represents   a   confocal  hyperbola   when    a^>k'^>b^;   given 
a>6. 

Determine  the  nature  of  the  following  conies;  and  also  their  foci, 
directrices,  centers,  semi-axes,  and  latera  recta : 

19.  ?/2  =  (x  +  3)  (x  +  4)  ; 

20.  x^-4:y'^  +  x  +  y  +  1=0', 

21.  x^  =  4:X  +  ny  +  7; 

22.  3x2  +  / _  6a; +  8?/ +  1=0; 

23.  3  x2  +  5  y  =  3  ?/2  +  5  a: ; 

24.  9(x^-y)  =-dy(l+2x-S y). 

25.  Show  that  the  polar  equation  of  the  parabola,  with  its  vertex  at  the 
,     .  4»cos^ 

26.  Show  that  if  the  left  hand  focus  be  taken  as  pole,  the  polar  equation 

of  the  ellipse  is  p  =  — ^^ ^• 

1  —  e  cos  d 

27.  Derive  the  polar  equation  of  an  hyperbola,  with  its  pole  at  the 
focus,  eccentricity  2,  and  the  distance  of  the  focus  from  the  directrix 
equal  to  6. 


CHAPTER   IX 
THE    PARABOLA  2/2=4  i)x 

133.  Review.  In  the  preceding  cliapter  (Arts.  102  to  108), 
the  nature  of  the  parabola  has  been  examined,  and  its  equa- 
tion derived  in  two  standard  forms.     These  equations  are  : 

^2_  4,px,  if  the  axis  of  the  curve  coincides  with  the  a;-axis, 
and  the  tangent  at  the  vertex  with  the  ^-axis ;  and 

(y  —  ^)2  =  4^  (a;  —  A),  if  the  axis  of  the  curve  is  parallel 
to  the  a;-axis,  and  the  vertex  is  at  the  point  (A,  ^).  In  the 
present  chapter,  some  of  the  intrinsic  properties  of  the  parab- 
ola are  to  be  studied,  i.e.,  properties  which  belong  to  the 
curve  and  are  entirely  independent  of  the  position  of  the 
coordinate  axes.  For  this  purpose,  it  will,  in  general,  be 
easier  to  use  the  simplest  form  of  the  equation  of  the  curve, 
viz.,  2/^  =  4:px. 

In  every  parabola,  the  value  of  the  eccentricity  is  e  =  1. 
If  the  equation  of  the  parabola  is  ^/^  =  4:px,  then  the  focus 
is  the  point  (jc»,  0),  the  directrix  is  the  line  x  =  —p,  and 
the  axis  of  the  curve  is  the  line  ?/  =  0.     The  equation 

y^y=2p(x  +  x^^ 

represents  the  polar  of  the  point  P^  =  (a^j,  y^)  with  respect 
to  the  parabola,  for  all  positions  of  P^.  If  P^  be  outside 
the  curve,  this  polar  is  the  chord  of  contact  corresj)onding 
to  tangents  from  P^ ;  if  P^  be  upon  the  curve,  this  polar 
is  the   tangent   at   that   point.     These  facts,  shown  in  the 

219 


220 


ANA  L  YTIC  GEOMETR  T 


[Ch.  iX. 


previous   chapter,    will   be    assumed   in   the    following    dis- 
cussion. 

134=  Construction  of  the  parabola.  The  two  conceptions 
of  a  locus  given  in  Article  35  lead  to  tAvo  methods  for  con- 
structing a  curve,  viz.,  by  plotting  points  to  be  connected 
by  a  smooth  curve,  and  by  the  motion  of  a  point  constrained 
by  some  mechanical  device  to  satisfy  the  law  which  defines 
the  curve.  These  two  methods  may  be  used  in  constructing 
a  parabola. 

(«)  By  separate  points.  Given  the  focus  F  and  the  vertex 
(7,  draw  the  axis  OFX,  the  directrix  D'D  cutting  this  axis 

in  Z^  and  also  a  series  of  lines 
perpendicular  to  the  axis  at 
J/^,  ilfg,  ilfg,  etc.,  respectively. 
With  F  as  center  and  ZM^ 
as  radius,  describe  arcs  cut- 
ting the  line  at  M^  in  two 
points  P^  and  Q^ ;  similarly, 
with  F  as  center  and  ZM^^  as 
radius,  cut  the  line  at  M^  in 
P^  and  §2 ;  ^^^  so  on.  The 
points  thus  found  evidently 
satisfy  the  definition  of  the  parabola  (Art.  102).  In  this 
Avay,  as  many  points  of  the  curve  as  are  desired  may  be 
found.  If  these  be  then  connected  by  a  smooth  curve,  it 
will  be  approximately  the  required  parabola  (cf.  Note  B, 
Appendix). 

(/3)  By  a  continuously  moving  point.  Let  D'B  be  the 
directrix  and  F  the  focus.  Place  a  right  triangle  with 
its  longer  side  KH  in  coincidence  with  the  axis  of  the 
curve,  and  its  shorter  side  KJ  in  coincidence  with  the  direc- 
trix.    Let  one  end  of  a  string  of  length  KH  be  fastened  at 


133-135.] 


THE  PARABOLA 


221 


jff",  and  the  other  end  at  F.  If 
now  a  pencil  point  be  pressed 
against  the  string,  keeping  it 
taut  while  the  triangle  is 
moved  along  the  directrix,  as 
indicated  in  the  figure,  then, 
in  every  position  of  P, 

FF  =  KP, 

therefore  the  pencil  will  trace 

an  arc  of  a  parabola.  ^  ^^^ 

135.   The  equation  of  the  tangent  to  the  parabola  y^  =  ^px 

in  terms  of   its  slope.     The  equation   of  a  line  having  the 

given  slope  m  is 

y  =  mx  -\-h\       .         »         o         (1) 

it  is  desired  to  find  that  value  of  k  for  which  this  line  will 
become  tangent  to  the  parabola  whose  equation  is 

y'^  =  4:px.  o  o  .  (2) 

Considering  equations  (1)  and  (2)  as  simultaneous,  and 
eliminating  y^  the  resulting  equation,  which  is 

(mx -\'ky^=4:px^  .  o  „  (3) 

has  for  its  roots  the  abscissas  of  the  two  points  in  which  the 
loci  of  equations  (1)  and  (2)  intersect.  These  roots  will 
become  equal  (cf.  Art,  9),  and  therefore  the  points  of  inter- 
section will  become  coincident,  if 

(mk  ~^ff  —  Tf^l^ 


0, 


^.g.,  if 
Therefore 


A-  = 


^V 


m 


y  =  tnoc  + 


P 
in 


(4) 

[56] 


is,  for   all  values    of  w,  the   equation  of   a  tangent  to  the 
parabola  ^2  _  4  ^^^ 


222 


ANALYTIC  GEOMETRY 


[Ch.  IX. 


The  abscissa  of  tlie  point  of  contact  of  the  loci  of  equa- 
tions (2)  and  \J)^^  may  be  found  from  equation  (3),  by  sub- 
stituting in  it  the  value  of  k  given  in  equation  (4);  it  is  i^. 


wr 


The  ordinate  may  then  be  found  from  equation  (1);  it  is 
—S..     The  point  of  contact  is  then  ( -^,  —^\ 

136.   The  equation  of  the  normal  to  the  parabola  y^  =  ^pqc 

in  terms  of  its  slope.  Since,  by  definition,  the  normal  to  a 
curve  is  perjDcndicular  to  the  tangent  at  the  point  of  con- 
tact, the  equation  of  a  normal  to  the  parabola 

y^  =  -ipx  .  .  .  (1) 

is,  if  m'  be  the  slope  of  the  tangent  [Arts.  62,  135], 

If  m  be  the  slope  of  the  normal,  then 

1 

m  = „ 

m 

and  equation  (2)  may  be  written 

y  =  mx  —  2pm  —  pm^.        .      .       .       [57] 
This  is  the  equation  of    a  normal   in   terms    of    its   own 

slope  m. 

137.   Subtangent   and 

subnormal.  Construction 

of  tangent  and  normal. 

Let  Pj  =  (x^,  y^  be 
any  given  point  on  the 
parabola  whose  equa- 
tion is 

^2—.  i^px.   .  .  .   (1) 

Draw    the     ordinate 

MP^,  the  tangent  ^P^ 

and    the    normal   P^^JV. 


Fig.  101 


135-137.]  THE  PARABOLA  223 

Then  by  the  definitions  of  Art.  86,  the  subtangent  is  TM^  the 
subnormal  is  MN^  the  tangent  length  TPy,  and  the  normal 
length  P^N,     The  tangent  at  P^  has  for  its  equation  (Art. 

"'^'  «/i^  =  2|?(x  +  a;i),      ...        (2) 

hence  its  a;-intercept  is  AT  =  —x-^.     But  AM=x-^^ 
therefore  TM=2x^. 

This  proves  that  the  subtangent  of  the  parabola  y^  =  4:px 
is  bisected  at  the  vertex;  and  that  its  length  is  equal  to  ttvice 
the  abscissa  of  the  point  of  contact. 

The  normal  at  P^  has  for  its  equation  (Art.  123) 

y-yi  =  -^Q^-^i)->     •      •      •     (3) 

hence  its  a;-intercept  is  AN=  x^-{-  2p.     But  AM=  x-^^ 
therefore  M]S/'=  2p. 

That  is,  in  words,  the  subnormal  of  the  jjarabola  y^  —  \px 
is  constant;  it  is  equal  to  half  the  latus  rectum. 

These  properties  of  the  subtangent  and  subnormal  give 
two  simple  methods  of  constructing  the  tangent  and  normal 
to  any  parabola  at  a  given  point,  if  the  axis  of  the  parabola 
is  given. 

First  method  :  from  the  given  point,  let  fall  a  perpendicu 
lar  P-JKto  the  axis  of  the  parabola,  meeting  it  in  M.  The 
vertex  of  the  curve  being  at  A^  construct  the  point  T  on  the 
axis  produced,  so  that  TA  =  AM.  The  straight  line  TP^  is 
the  required  tangent  at  P^,  and  a  line  through  P^  at  right 
angles  to  this  tangent  is  the  required  normal. 

Second  method :  from  the  foot  of  the  perpendicular  MP^ 
construct  the  point  iV,  so  that  MN  equals  twice  the  distance 
from  vertex  to  the  focus  (2  j?  =  2  AF^  ;  then  P^N  is  the 
required  normal,  and  a  line  through  P^  at  right  angles  to 
P^N  is  the  required  tangent. 


224  ANALYTIC  GEOMETRY  [Ch.  IX. 

EXERCISES 

1.  Construct  a  parabola  with  focus  2^'^'^  from  the  dh'ectrix. 

2.  Construcfc  a  parabola  with  latus  rectum  equal  to  6. 

3.  Find  the  equations  of  the  two  tangents  to  the  parabola  y'^  =  4:px, 
which  form  with  the  tangent  at  the  vertex  a  circumscribed  equilateral 
triangle.  Find  also  the  ratio  of  the  area  of  this  triangle  to  the  area  of 
the  triangle  whose  vertices  are  the  points  of  tangency. 

4.  Find  the  equation  of  a  tangent  to  the  parabola  y'^  =  4jt?x,  perpen- 
dicular to  the  line  4  ?/  —  a;  +  3  =  0,  and  find  its  point  of  contact. 

5.  Find  the  equations  of  the  tw^o  tangents  to  the  parabola  y^  =  bx 
from  the  point  (7,  1),  using  formula  [56]. 

6.  Write  the  equations  of  the  tangents  to  the  parabola  y"^  =  10  x,  at 
the  extremities  of  the  latus  rectum.  On  what  line  do  these  tangents 
intersect?  (cf.  Art.  138  (5),  p.  228.) 

7.  Write  the  equations  of  the  tangent  and  normal  to  the  parabola 
2/2  —  9  a;^  at  the  point  (1,  6). 

8.  Write  the  equation  of  the  normal  to  the  parabola  y^  =  Qx,  drawn 
through  the  point  (|,  3). 

9.  Write  the  equation  of  the  tangent  to  the  parabola  y"^  =  ^px,  for 
the  point  for  which  the  normal  length  is  twice  the  subtangent;  for  the 
point  for  which  the  normal  length  is  equal  to  the  difference  between  the 
subtangent  and  subnormal. 

10.  Two  equal  parabolas  have  the  same  vertex,  and  their  axes  are  at 
right  angles ;  find  the  equation  of  their  common  tangent,  and  show  that 
the  points  of  contact  are  each  at  the  extremity  of  a  latus  rectum. 

11.  Find  the  locus  of  the  middle  point  of  the  normal  length  of  the 
parabola  y^  =  4jox. 

12.  The  subtangent  of  a  parabola  for  the  point  (5,  4)  is  10 ;  find  the 
equation  of  the  curve,  and  length  of  the  subnormal. 

13.  Find  the  subtangent,  and  the  normal  length,  for  the  point  whose 
abscissa  =  —  6,  and  w4iich  is  on  the  parabola  ?/2  =  —  6x. 

14.  Find  the  equation  of  the  tangent  parallel  to  the  polar  of  ("1,  2) 
with  respect  to  the  parabola  y'^  =  \2x\  also  find  the  point  of  contact, 
the  length  of  the  tangent,  and  the  subtangent. 

15.  Find  the  equation  of  a  parabola  which  is  tangent  to  the  line 
2y  —  ?>x  —  1,  and  whose  axis  is  parallel  to  the  x-axis. 


137-138.] 


THE  PARABOLA 


225 


16.  Show  that  the  sum  of  the  subtangent  and  subnormal  for  any 
point  on  the  parabola  y'^  =  4:px,  equals  one  half  the  length  of  focal  chord 
parallel  to  the  corresponding  tangent. 

17.  Show  that  as  the  abscissa  in  the  parabola  y'^  =  ^px  increases 
from  0  to  CO,  the  slope  of  the  tangent  diminishes  from  go  to  0;  hence  the 
curve  is  concave  toward  its  axis. 

< 

138.  Some  properties  of  the  parabola  which  involve  tangents 
and  normals.     Let  F  be  the  focus,  A  the  vertex,  AX  the 


Fig.  103. 


axis,  and  i)'2>  the  directrix  of  the  parabola  whose  equation  is 


y^  =  4lpx. 


(1) 


Through  any  point  P^  =  (xyf  y^)  on  the  curve  draw  the 
tangent  TP^^  cutting  the  ?/-axis  in  R^  the  directrix  in  S^  and 
the  a:-axis  in  T\  also  draw  the  normal  P^N],  the  focal  chord 
P^FP^ ;  the  tangent  at  P^ ;  the  lines  L^P^  Q  and  L^P^,  per- 
pendicular to  the  directrix ;  and  the  lines  SF  and  L^F. 
Then  the  following  properties  of  the  parabola  are  readily 
obtained : 


TAN.  AN.  GEOM. 


15 


226  AyALYTIC   GEOMETRY  [Ch.  IX. 

(1)  The  focus  is  equidistant  from  the  points  P^,  T^  and  N. 
For  ZPj  =  XjPi  =  ZA  +  AM^  =  x^+p, 

TF=TA  +  AF=x^-\-  p,  Art.  137 

and  F]Sr=  AM^  +  {M^N -  AF)  =  x^-hp;    Art.  137 

hence  FP^=TF=FK 

The  point  F  is  the  midpoint  of  the  hypotenuse  of  the 
right  triangle  TP^N,  and  is  therefore  equidistant  from  the 
vertices  T^  P^,  and  N.  Thus  a  third  method  is  suggested  for 
constructing  the  tangent  and  normal  at  Pj,  viz. :  by  means  of 
a  circle,  with  the  focus  F  as  center,  and  the  focal  radius  FP^ 
as  radius,  which  cuts  the  axis  in  T  and  N. 

(2)  The  tange^it  and  normal  bisect  internally  and  externally^ 
respectively^  the  angle  between  the  focal  radius  to  the  point  of 
contact  ayid  the  perpendicular  from  that  point  to  the  directrix. 

For  ZL^P^T=ZP^TF,  since  L^P^  \\  TF-, 

and  Z  TP^F  =  ZP^  TF,  since  TF  =  FP^ ; 

ZL^P^T  =  Z.TP^F. 
Also,         Z.FP^N=Z.NP^Q,  since  P^N'±P^T. 

(3)  Tlirough  any  point  in  the  plane  two  taiigents  can  be 
dratvn  to  the  parabola  (cf.  Arts.  89,  125). 

The  line  y  =  mx  +  ^ (1) 

m 

is  tangent  to  the  parabola  y'^  =  Apx  for  all  values  of  m.  If 
P'  =  (x\  y'^  be  any  given  point  of  the  plane,  then  the  tan- 
gent (1)  will  pass  through  P'  if,  and  only  if,  m  satisfy  the 

equation  ,  ,      v 

y'  =  mx   +  =^, 
m 


i.e.,ii  n^l±2tt:=A^.       .       .       .       (2) 

Therefore  two,  and  only  two,  values  of  m  satisfy  the  given 
conditions;  and  therefore  through  any  point  of  the  plane  two 


138.]  THE  PARABOLA  227 

tangents  can  be  drawn  to  the  parabola.  If,  however,  P'  is 
on  the  curve,  then  y'^  -  ipx'  =  0,  the  two  values  of  m  are 
equal,  i.e.,  the  two  tangents  coincide.  If  F'  is  inside  the 
parabola,  then  ?/'"^  —  4  px'  <  0,  and  the  two  values  of  m  are 
imaginary,  i.e.,  there  are  no  real  tangent  lines.  Therefore 
it  is  only  when  F'  is  outside  the  parabola  that  two  real  and 
different  tangent  lines  may  be  drawn  from  it  to  the  parabola. 

(4)  Thi'ougli  any  point  in  the  plane  three  normals  can  he 
drawn  to  the  parabola. 

The  line  y  =  mx—2pm  —  pm^       .         .         .         (1) 

is  normal  to  the  parabola  y'^  =  4:  px  for  all  values  of  m 
(Art.  136).  If  F'  =  (x',  y')  be  any  point  of  the  plane,  then 
the  normal  (1)  will  pass  through  P'  if,  and  only  if,  m  has  a 
value  that  will  satisfy  the  equation 

.  y'  =  x'm  —  2  pm  —  pm^.  .  .  .  (2) 
Since  equation  (2)  is  a  cubic  in  m,  there  are  three  values  of 
m  which  satisfy  the  given  conditions,  and  therefore,  in  gen- 
eral, three  normals  may  be  drawn  to  a  parabola  from  a  given 
point.  Special  cases  may,  however,  arise  in  which  two  of 
the  roots  of  equation  (2)  are  equal,  when  there  would  be 
only  two  different  normal  lines ;  or  all  the  roots  may  be 
equal,*  or  two  imaginary  and  one  real,  in  both  of  which 
cases  there  would  be  only  one  normal  line.  Through  every 
point  at  least  one  normal  line  can  be  drawn  to  the  parabola. 

(5)  The  tangents  at  the  extremities  of  a  focal  chord  intersect 
on  the  directrix,  and  at  7'ight  angles  (cf.  (6),  below). 

For,  if  S=  (x\  y')  is  the  point  of  intersection  of  the 
tangents  at  the  extremities  of  the  focal  chord,  then  the  chord 
is  the  polar  of  aS^,  and  its  equation  is 

y'y=2p(x  +  x^).        .         .        .         (1) 

*  For  only  one  point,  viz. :  P'  =  (2p,  0),  are  all  the  roots  of  equation  (2)  equal. 


228  ANALYTIC  GEOMETRY  [Ch.  IX. 

But  since  this  line  passes  through  the  focus  F={p,  0), 

0  =  2^:>(^:)  H-  x')  ; 

i.e.,  x'  =  —  p.  .  .  .  (2) 

Hence  the  point  P'  is  on  the  locus  x=—p,  i.e.,  on  the 
directrix. 

Again,  the  tangent  line 

passes  through  the  point     P'  =[  —  p,  y') 

if  y^  —  —  mp  +  — ' 

i.e.,  if  vi^ +  —  771  —  1  =  0.       .        .        .        C4) 

p  ^ 

But  the  roots  of  equation  (4)  are  the  slopes  m'  and  m"  of 
the  two  tangents  at  P^  and  P^ ;  and  by  Art.  11, 

m'm"  =  —  1. 

Hence,  the  tangents  at  Pi  and  Pg  intersect  at  right  angles. 

(6)  The  line  joiniyig  any  point  in  the  directrix  to  the  focus 
of  a  pardhola  is  p)erj)endicular  to  the  chord  of  contact  cor- 
responding to  that  point. 

For  ASL,P,  =  ASFP, 

since      L^Pi  =  FP^,  SP^  is  common,  Z  L,P^S=  Z.  SP^F; 

hence,  Z  SFP,  =  Z  SL,P,  =  90°. 

The  property  of  (5)  may  now  be  shown  geometrically. 
Draw  the  tangent  at  P2,  and  suppose  it  to  meet  the  directrix 
in  aS"  ;  then,  by  what  has  just  been  proved,  Z  S' FP2  is  a 
right  angle  ;  then  FS'  must  coincide  with  FS ;  and  the  tan- 
gents at  Pi  and  P^  meet  on  the  directrix. 


138.]  THE  PAUABOLA  229 

Moreover,  Z  F2SF1  is  11  right  angle,  for  SPi  bisects 
Z  FjSL,,  and  SF^  bisects  Z  L,SF. ' 

(7)  J.  perpendicular  let  fall  from  the  focus  upon  a  tangent 
line  meets  that  tangent  upon  the  tangent  at  the  vertex. 

For  the  equation  of  the  tangent  at  Fi  is 

1/ig  =  2px  -^  ^pxi,        .        .        ,        (1) 

and  the   equation  of  the  perpendicular  through  the  focus 
F  =  (p,  0)  is 

2pg  ==  -  y^xArpVv         •         •        •'        (2) 

Regarding  equations  (1)  and  (2)  as  simultaneous,  and 
solving  to  find  the  point  of  intersection  i?,  its  abscissa  is 
determined  by  the  equation 

(4^2  _^  y2^^  j^  p(4:pxi  -  yf^  =  0  ; 
or,  since  g^^  =  4:pxi^ 

x=0;      .     , (3) 

and  M  is  therefore  on  the  tangent  at  ^. 

JSToTE.  The  preceding  properties  of  the  parabola  have  for  variety 
been  given  in  some  cases  a  geometric,  in  others  an  analytic,  proof.  The 
student  is  advised  to  use  both  methods  of  proof  for  each  proposition. 
Other  properties  of  the  parabola  are  given  below  as  exercises  for  the 
student,  and  should  be  derived  by  analytic  methods. 

EXERCISES 

1.  Write  the  equations  of  the  normals  drawn  through  the  point  (3,  3) 
to  the  parabola  y"^  =  Q  x  . 

2.  The  focal  distance  of  any  point  of  the  parabola  2/^  =  4:px  is  p  -\-  x. 

3.  The  circle  on  a  focal  chord  as  diameter  touches  the  directrix. 

4.  The  angle  between  two  tangents  to  a  parabola  is  one  half  the 
angle  between  the  focal  radii  of  the  points  of  tangency. 


230  ANALYTIC  GEOMETRY  [Ch.  IX. 

5.  The  polars  of  all  points  on  the  latus  rectum  meet  the  axis  of  the 
parabola   in   the   same   point ;    find    its   coordinates,   for   the   parabola 

6.  The  product  of  the  segments  of  any  focal  chord  of  the  parabola 
y^  =  4:px  equals  7^  times  the  length  of  the  chord. 

7.  Two  tangents  are  drawn  from  an  external  point  P.^^  =  (x^,  y^  to 
a  parabola,  and  a  third  is  drawn  parallel  to  their  chord  of  contact.  The 
intersections  of  the  third  with  each  of  the  other  two  is  half  way  between 
Pj  and  the  corresponding  point  of  contact. 

8.  The  area  of  a  triangle  formed  by  three  tangents  to  a  parabola  is 
one  half  the  area  of  the  triangle  formed  by  the  three  points  of  tangency. 

9.  The  tangent  at  any  point  of  the  parabola  will  meet  the  directrix 
and  latus  rectum  produced,  in  two  points  equidistant  from  the  focus. 

10.  The  normal  at  one  extremity  of  the  latus  rectum  of  a  parabola  is 
parallel  to  the  tangent  at  the  other  extremity. 

11.  The  tangents  at  the  ends  of  the  latus  rectum  are  twice  as  far 
from  the  focus  as  they  are  from  the  vertex. 

12.  The  circle  on  any  focal  radius  as  diameter  touches  the  tangent 
drawn  at  the  vertex  of  the  parabola. 

13.  The  line  joining  the  focus  to  the  pole  of  a  chord  bisects  the  angle 
subtended  at  the  focus  by  the  chord. 

14.  Prove,  geometrically,  that  a  perpendicular  let  fall  from  the  focus 
upon  a  tangent  line  of  a  parabola  meets  that  tangent  upon  the  tangent 
drawn  at  the  vertex  (cf .  (7)  of  Art.  138,  p.  229). 

139.  Diameters.  A  diameter  lias  been  defined  as  the 
locus  of  the  middle  points  of  a  S3^stem  of  parallel  chords. 
Its  equation  may  be  found  as  follows  (cf.  Art.  129): 

Let  m  be  the  common  slope  of  a  system  of  parallel  chords 
of  the  parabola  whose  equation  is 

^2  =  4^3^,  .         .  .  (1) 

then  the  equation  of  one  of  these  chords  is 

y  z=  mx  H-  ^,  .         .         V-         (2) 


138-139.] 


THE  PARABOLA 


231 


and  the  equation  of  any  other  chord  of  the  system  will  differ 
from  this  only  in  the  value 
of    the    constant    term   k. 
The  chord  (2)  meets  the 
parabola  (1)  in  two  points 

and       P^  =  (x^,y^, 

and  the  coordinates  of  the 
middle  point  P'  =  (x' ^  y') 
are  therefore 


Fig. 103. 


^f^^^i+^,  and  y^  =  y±±^.  . 


2 


(3) 


Considering  (1)  and  (2)  as  simultaneous  equations,  and 
eliminating  x^  it  follows  that  the  ordinates  of  P^  and  P^  are 
the  roots  of  the  equation 

my'^  =  4  j?(y  —  ^), 


i.e.^  of 


y — ^y  +  -^— =  0. 
m  m 


(4) 


Therefore,  by  Art.  11, 


^1  +  ^2  =  -^'  ^•^•'  y  =~-^ 

m  m 

hence  whatever  the  value  of  ^,  the  coordinates  of  the  middle 
point  of  the  chord  satisfy  the  equation 


2p 
y=^' 
m 


(6) 


This  is,  therefore,  the  equation  of  the  diameter  correspond- 
ing to  the  system  of  chords  whose  slope  is  w.* 

*  Equation  (5)  might  have  been  obtained  at  once  as  a  special  form  of 
equation  [54],  Art.  129,  by  giving  appropriate  values  to  the  coefficients  A,  J5, 
F,  G,  and  C  there  used. 


232  ANALTTIC  GEOMETRY  [Ch.  IX. 

140.   Some  properties  of  the  parabola  involving  diameters. 

The  equation  of  the  diameter  of  the  parabola  (Art.  139), 

y-"^,      ...  (1) 

m 

shows  at  once  that  every  dimneter  of  the  parabola  is  parallel 
to  the  axis  of  the  curve.      (See  also  Ex.  8,  p.  213.) 

Conversely,  since  any  value  whatever  may  be  assigned 
to  m,  each  value  deterniining  a  system  of  parallel  chords, 
equation  (1)  may  represent  any  line  parallel  to  the  aj-axis, 
and  therefore  every  line  parallel  to  the  axis  of  a  parabola  bisects 
some  set  of  parallel  chords^  and  is  a  diameter  of  the  curve. 

Again,  each  of  the  chords  cuts  the  parabola  in  general  in 
two  distinct  points,  and  the  nearer  these  chords  are  to  the 
extremity  of  the  diameter  the  nearer  are  these  two  points 
to  each  other  and  to  their  mid-point.  In  the  limiting  posi- 
tion, when  the  chord  passes  through  the  extremity  of  the 
diameter,  the  two  intersection  points  and  their  mid-point 
become  coincident,  and  the  chord  is  a  tangent.  Therefore 
the  tangent  at  the  end  of  a  diameter  is  parallel  to  the  bisected 
chords. 

It  follows  from  the  preceding  properties,  or  directly  from 
equation  (1),  that  the  axis  of  the  parabola  is  the  only  diameter 
perpendicidar  to  the  tangent  at  its  extremity. 

The  student  will  readily  perceive  how  the  above  properties 
give  a  method  for  constructing  a  diameter  to  a  set  of  chords, 
and  in  particular  how  to  construct  the  axis  of  a  given  parab- 
ola. Thus  the  problem  of  Art.  137,  to  construct  a  tangent 
and  normal  to  a  given  parabola  at  a  given  point,  can  now  be 
solved  even  when  the  axis  is  not  given. 

If  any  point  on  a  diameter  is  taken  as  a  pole,  its  polar 
will  be  one  of  the  system  of  bisected  chords,  of   slope  m. 


140-141.] 


THE  PAEABOLA 


233 


For  the  pole  isP'  =  [x' ^  —  ),  hence  the  equation  of  its  polar 

(Art  127)  is 

2  p 


t.e. 


y  =  mx  +  mx\ 

which  is  the  equation  of  a  chord  of  slope  m.  In  other  words, 
the  tangents  at  the  extremities  of  a  chord  of  the  parabola  inter- 
sect upon  the  corresponding  diameter. 


141.    The   equation  of   a  parabola  referred   to   any  diameter  and  the 
tangent  at  its  extremity  as  axes.     In  the  simplest  form  of  the  equation 

of  the  parabola,  viz., 

y^  =  ^px,  .  .  .  (1) 

the  coordinate  axes  are  the  principal  diameter  and  the  tangent  at  its 
extremity.  These  are  the  only  pair  of  such  lines  that  are  perpendicular 
to  each  other  (Art.  140).  It  is  now  desired  to  find  the  equation  of  the 
parabola,  when  referred  to  any  diameter  of  the  curve  and  the  tangent  at 
its  extremity  as  axes. 

Let  any  diameter  O'X'  of  the  parabola  (1)  be  the  new  3:-axis,  and  the 
tangent   O'Y'  at   0'   be   the   new 
?/-axis,  meeting  the  old  a:-axis  at 
an  angle  6. 

If  m  =  tan  0,     .     .     .     (2) 

then   (Art.    135)   the   coordinates 

of   0'   are  -^    and     — ,  and  the 
m^  m 

equation  for  transforming  the 
equation  from  the  old  axes  to  a 
parallel  set  through  the  point  0' 
are  (Art.  71),  Fio.lOi. 


^  =  ^''+£-2'  2/  =  /  + 


m 


2p 

m 


Substituting  these  values  in  equation  (1)  gives 

y2  +  ^y^4^^^ 


(^) 


(4) 


234  ANALYTIC  GEOMETRY  [Ch.  IX. 

To  turn  the  y-axis  to  the  final  position,  making  an  angle  $  with  the 
a:-axis,  the  equations  for  transformation  are  (Art.  72,  [24]), 

x'  =  x"  +  y"  cos  6,  y'  =  y"  sin  6, 
or,  by  equation  (2), 

x'  =  x"  +  —~=,     and     y'  =     J^^  o      .      .      (5) 

V 1  +  mP-  Vl  +  w2 

Substituting  these  values  in  equation  (4),  it  becomes 


m^ 


-y  2  =  ^p^n . 


1  +  w^' 
or,  dropping  now  the  accents, 

which  is  the  required  equation  of  the  parabola. 

This  equation   may,  however,  be  written  more  simply.     Observing 

(Art.  103)  that  JO  (  —^ —  1  is  the  focal  distance  of  the  new  origm  0',  and 

representing  that  distance  by  p',  equation  (6)  becomes 

?/2  =  4y:r.  .  .  .  [58] 

This  equation  is  of  the  same  form  as  equation  (1),  but  is  referred  to 
oblique  axes.     In  general,  therefore,  the  equation 

represents  a  parabola,  and  -  is  the  distance  of  its  focus  from  the  origin. 

Equation  [58]  states  the  following  property  for  every  point  P  of  the 
parabola  • 

MP^  =  4:F0''0'M  ] 

a  property  entirely  analogous  to  that  of  Arte  106. 

EXERCISES 

1.  Find  the  diameter  of  y^=—7x,  which  bisects  the  chords  parallel 

to  the  line  x  —  y  +  2  =  0. 

2.  A  diameter  of  the   parabola  y^  =  8x  passes  through  the  point 
(2,  -3)  ;  what  is  the  equation  of  its  corresponding  chords  ? 

3.  Find  the   equation  of   the  diameter  of  the  parabola  ?/2  =  4  a;  -f  4 
which  bisects  the  chords  2y  —  3x  =  k. 

4.  Find  the  equation  of  the  tangent  to  the  parabola  (y  —  6)2=8(a;+2), 
which  is  perpendicular  to  the  diameter  ?/  —  4  =  0. 


141.]  THE  PARABOLA  ^  235 

5.  Show  that  the  pole  of  any  chord  is  on  the  diameter  which  corre- 
sponds to  the  chord. 

6.  What  is  the  equation  of  the  parabola  y^  =  Sx,  when  referred 
to  its  diameter  y  —  o  =  0  and  the  corresponding  tangent  as  coordinate 
axes? 

7.  What  is  the  equation  of  the  parabola  (x  +  3)^  =  12  (y  —  1), 
when  referred  to  a  diameter  through  the  point  (3,  4)  and  the  corre- 
sponding tangent  as  coordinate  axes? 

8.  Find  the  pole  of  the  diameter  y  =  k  with  reference  to  the  parab- 
ola y^  =  4:px. 

9.  The  polar  of  any  point  on  a  diameter  is  parallel  to  the  correspond- 
ing tangent  of  that  diameter. 

EXAMPLES   ON    CHAPTER    IX 
Find  the  equation  of  a  parabola  with  axis  parallel  to  the  a:-axis : 

1.  passing  through  the  points  (0,  0),  (3,  2),  (3,  -2) ; 

2.  passing  through  the  points  (1,  1),  ("3,  -3),  (2,  2) ; 

3.  through  the  point  (4,  "5),  with  the  vertex  at  the  point  (3,  -7). 

4.  Find  the  equation  of  a  parabola  through  the  four  points  (0,  2), 
(3,0),  (-1,-1),  (-3,-2). 

5.  Find  the  vertex  and  axis  of  the  parabola  of  Ex.  4. 

Find  the  equation  of  a  parabola 

6.  if  the  axis  and  directrix  are  taken  as  coordinate  axes. 

7.  with  the  focus  at  the  origin,  and  the  ?/-axis  parallel  to  the  directrix. 

8.  tangent  to  the  line  4:y  =  Sx  —  12,  the  equation  being  in  the  sim- 
plest standard  form. 

9.  if  a  focal  radius  of  length  10  lies  along  the  line  4a;  —  3?/  —  8  =  0. 

10.  Two  equal  parabolas  have  the  same  vertex,  and  their  axes  are  per- 
pendicular ;  find  their  common  chord  and  common  tangent  (cf .  Ex.  10, 
p.  224). 

11.  At  what  angle  do  the  parabolas  of  Ex.  10  intersect? 

12.  Two  tangents  to  a  parabola  are  perpendicular  to  each  other ;  find 
the  product  of  the  corresponding  sub-tangents. 

Find  the  locus  of  the  middle  point 

13.  of  all  the  ordinates  of  a  parabola. 

14.  of  all  chords  passing  through  the  vertex. 


236  ANALYTIC  GEOMETRY  [Ch.  IX.  141. 

15.  From  any  point  on  the  latus  rectum  of  a  parabola,  perpendiculars 
are  drawn  to  the  tangents  at  its  extremities ;  show  that  the  line  joining 
the  feet  of  these  perpendiculars  is  a  tangent  to  the  parabola. 

16.  If  tangents  are  drawn  to  the  parabola  ^/^  =  4  ax  from  any  point 
on  the  line  a;  +  4fl  =  0,  their  chord  of  contact  will  subtend  a  right  angle 
at  the  vertex. 

Two  tangents  of  slope  m  and  m',  respectively,  are  drawn  to  a  parab- 
ola ;  find  the  locus  of  their  intersection : 


17. 

if  mm'  =  k', 

18. 

if  -  +  —  =  A' ; 
111      m 

19. 

m      m' 

20.  Find  the  locus  of  the  center  of  a  circle  which  passes  through  a 
given  point,  and  touches  a  given  line. 

21.  The  latus  rectum  of  the  parabola  is  a  third  proportional  to  any 
abscissa  and  the  corresponding  ordinate. 

22.  Find  the  locus  of  the  point  of  intersection  of  tangents  drawn  at 
points  whose  ordinates  are  in  a  constant  ratio. 

23.  What  is  the  equation  of  the  chord  of  the  parabola  y^  =  Sx  whose 
middle  point  is  at  (2,  -5)  ? 

24.  A  double  ordinate  of  the  parabola  y^  =  4opx  is  8/?;  prove  that 
the  lines  from  the  vertex  to  its  two  ends  are  perpendicular  to  each  other. 

25.  Find  the  locus  of  the  center  of  a  circle  which  is  tangent  to  a  given 
circle  and  also  to  a  given  straight  line. 

26.  Find  the  intersections  of  a  normal  to  the  parabola  wdth  the  curve, 
and  the  length  of  the  intercepted  portion. 

27.  Prove  that  the  locus  of  the  middle  point  of  the  normal  intercepted 
between  the  parabola  and  its  axis  is  a  parabola  whose  vertex  is  the  focus, 
and  whose  latus  rectum  is  one  fourth  that  of  the  original  parabola. 

28.  Prove  that  two  confocal  parabolas,  with  their  axes  in  opposite 
directions,  intersect  at  right  angles. 

29.  Find  the  equation  of  the  parabola  when  referred  to  tangents 
at  the  extremities  of  the  latus  rectum  as  coordinate  axes. 

30.  The  product  of  the  tangent  and  normal  lengths  for  a  certain  point 
of  the  parabola  y^  =  4:px  is  twice  the  square  of  the  corresponding  ordi- 
nate ;  find  the  point  and  the  slope  of  the  tangent  line. 


CHAPTER  X 


THE    ELLIPSE,  ^^  +  f-  =  1 

142.  Review.  In  Chapter  VIII  the  nature  of  the  ellipse 
has  been  briefly  discussed,  and  its  equation  found  in  the  two 
standard  forms : 

—  +  ^  =  1, 

when  the  axes  of  the  curve  are  coincident  with  the  coordi- 
nate axes ;  and 

a?        "^        ^2        --L' 

Avhen  the  axes  of  the  curve  are  parallel  to  the  coordinate 
axes,  and  the  center  is  the  point  (A,  ¥).  In  the  present 
chapter  it  is  desired  to  study  some  of  the  intrinsic  properties 
of  the  ellipse,  ^.e.,  properties  which  belong  to  the  curve  but 
are  independent  of  the  coordinate  axes  ;  and  these  can  for  the 
most  part  be  obtained  most  easily  from  the  simpler  equation, 

^4-^  =  1 

The  ellipse  —  + 1-  =  1  has  its  eccentricity  given  bv  the 

^2  _  52 

relation   h"^  =  0^(1  —  e^')^    i.e.,   e^  = — ;    its   foci   are   the 

two  points  (±ae,  0),  and  its  directrices  the  lines  x  =  ±^ 

e 
(Art.  110).     If  the  axes  are  equal,  so  that  b  =  a,  the  curve 

takes  the  special  form  of  the  circle,  with  eccentricity  e  =  0, 

237 


238  ANALYTIC   GEOMETRY  [Ch.  X. 

the  two  foci  coincident  at  the  center,  and  the  directrices 
infinitely  distant. 

Tlie   equation  ^  +  ^^  =  1    represents  the  polar  of   the 

point  (a;^,  y-^  with  respect  to  the  ellipse ;  if  the  point  is 
outside  the  curve,  this  polar  line  is  its  chord  of  contact ; 
if  upon  the  curve,  the  polar  is  the  tangent  at  that  point 
(Arts.  122,  126,  127). 

These  facts  will  be  assumed  in  the  following  work. 

143.   The  equation  of  the  tangent  to  the  ellipse  ~^+    2=1 
in  terms  of  its  slope.     The  equation  of  a  line  having  the 

2^1  ven  slope  m  is  ,7  ^-,  ^ 

^  ^  y  =  mx  -\-  k  ;         .  .         .  (1) 

it  is  desired  to  find  that  value  of  k  for  which  this  line  will 
become  tangent  to  the  ellipse  whose  equation  is 

^  +  1  =  1.         ...         (2) 

Considering  equations  (1)  and  (2)  as  simultaneous,  and 
eliminating  y,  the  resulting  equation 

(b^  4-  a^m^^x^  +  2  ahnkx  +  aV  —  c^W'  =  0  .     .     (3) 

determines  the  abscissas  of  the  two  points  of  intersection  of 
the  curves  (1)  and  (2).  When  the  curves  are  tangent,  these 
abscissas  are  equal ;  therefore 


and  k  =  ±Vah7i^  -\-  b^. 


Hence  y  =  mx  ±  ^ cv^rr^  +  6^       .       .       .       [59] 

is  the  equation  of  a  tangent  to  the  ellipse  —  +  ^  =  1,  for  all 
values  of  m. 


142-144.] 


THE  ELLIPSE 


239 


Equation  [59]  shows  that  there  are  two  tangents  to  an 
ellipse  parallel  to  any  given  line ;  and  also  (Art.  125),  that 
there  are  two  tangents  to  an  ellipse  from  any  external  pomt. 

144.  The  sum  of  the  focal  distances  of  any  point  on  an 
ellipse  is  constant ;  it  is  equal  to  the  major  axis. 

2  2 

The  ellipse  ^  +  ^  =  1  has  its  foci  at  the  points 


a 


2  J2 


i^i  =  (  —  ae,  0)    and   F^  =  (ae,  0) ; 
with  52  =  «2  _  ^2^2^     ^cf^  ^j.^^  iiQ^ 

Let  Pi=(^Xi,  7/1)  be  any  point  on  the  curve,  so  that 


m'  =  ^'- 


hW 


\B' 
Fig. 105. 


Then,      F^P^  =  (x^  4-  ae^  +  ^1'  =  ah^  +  2  aex^  +  x^^  +  Vi^ 
=  a?e^  +  2  aex^  +  ^1'  +  ^'  -  ^ 


a^ 


I.e., 


=  aV  +  2  aex,  +  ^^'~  ^'^2^1^  +  a^-aV 

a" 

=  rt^  +  2  aea^i  +  ^'^Xi  ; 


240  AJS^ALYTIC   GEOMETRY  [Ch.  X. 


Again,     F2P1  =  (^1  —  ct^y  +  7/1  =  a^e^  —  2  aex^ + x^  +  y^^ 
=  a^  —  2  aexi  +  A^^ 

i.e.^  F2P1  ==  a  —  exy 

Hence,  by  addition, 

F,P,  4-  F,F,  =  2ai 

i.e.,  the  sum  of  the  focal  distances  of  any  'point  on  an  ellipse 
is  constant ;  it  is  equal  to  the  major  axis. 

This  property  gives  an  easy  method  of  finding  the  foci  of 
an  ellipse  when  the  axes  A' A  and  B'B  are  given. 

For  F^B  +  F,B  =  2a\ 

but  F,0  =  OF2, 

F2B  =  F,B=a. 

Hence,  to  find  the  foci,  describe  arcs  with  B  as  center  and 
a  =  OA  as  radius,  cutting  A' A  in  the  points  Fi  and  F2; 
these  points  are  the  required  foci. 

145.  Construction  of  the  ellipse.  The  property  of  Art. 
144  is  sometimes  given  as  the  definition  of  the  ellipse  ;  viz. 
the  ellipse  is  the  locus  of  a  point  the  sum  of  whose  distances 
from  two  fixed  points  is  constant.  This  definition  leads  at 
once  to  the  equation  of  the  curve  (cf.  Ex.  5,  p.  67);  and 
also  gives  a  ready  method  for  its  construction. 

(a)  Construction  hy  separate  points.  Let  A^ A  be  the 
given  sum  of  the  focal  distances,  i.e.,  the  major  axis  of  the 
ellipse  ;  and  Fy  and  F^  be  the  given  fixed  points,  the  foci. 
With  either  focus  as  center,  and  with  any  radius  A'R<A'A 
describe  an  arc  ;  then  with  the  other  focus  as  center,  and 
radius  RA^  describe  an  arc  cutting  the  first  arc  in  two 
points.     These  are  points  of  the  ellipse.     In  the  same  way 


144-145.]  THE  ELLIPSE  241 

as   many  points  as  desired  may  be  constructed;   a  smooth 
curve  connecting  these  points  is  approximately  an  ellipse. 


R A  F,  F, 

a  a »' 


'\ 

Fig, 106, 

(yS)  Construction  hy  a  continuously  moving  point.  Fix  two 
upright  pins  at  the  foci,  and  over  them  place  a  loop  of  string, 
equal  in  length  to  the  major  axis  plus  the  distance  between 
the  foci.  Press  a  pencil  point  against  the  chord  so  as  to 
keep  it  taut.  As  the  pencil  moves  around  the  foci,  it  will 
trace  an  ellipse. 

EXERCISES 

1.  Construct  an  eUipse  with  semi-axes  8^"^  and  6<=™. 

2.  Construct  an  ellipse  with  semi-axes  S''"^  and  12"^™. 

3.  Construct  an  ellipse  with  the  distance  between  the  foci  24,  and 
the  minor  axis  of  length  10. 

4.  Write  the  equation  of  the   polar  of   the  left-hand  focus  of  the 

2  2 

ellipse  ^  +  ^  =  1.     What  line  is  this  ? 

5.  By  employing  equation  [59],  find  the  tangent  to  the  ellipse 
16  x'^  +  2c  if  =  400,  and  passing  through  the  point  (3,  4). 

6.  By  the  method  of  Ex.  17,  p.  225,  show  that  an  ellipse  is  concave 
toward  its  center. 

7.  Throua:h  what  point  of  the  ellipse  —  +  ^  =  1  must  a  tangent  and 
normal  be  drawn,  to  form  with  the  a;-axis  an  isosceles  triangle  ? 

8.  Write  the  equations  of  the  tangent  and  normal  at  the  positive  end 
of  the  latus  rectum  of  the  ellipse  x^  +  4  2/2  =  4.  Where  do  these  lines  cut 
the  a;-axis? 

TAN.    AN.    GEOM. 16 


242  ANALYTIC   GEOMETRY  [Ch.  X. 

9.    Tangents  to  tlie  ellipse  4  x'^  -{-  3i/^  =  o  are  inclined  at  60°  to  the 
a;-axis ;  find  the  points  of  contact. 

10.  Find  the  equation  of  an  ellipse  (center  at  the  origin)  of  eccen- 
tricity f,  such  that  the  subtangent  for  the  point  (3,  -^^)  is  (—  -^/). 

11.  Find  the  chord  of  contact  for  tangents  from  the  point  (3,  2)  to 
the  ellipse  x^  +  4?/^  =  4.  Find  also  the  equation  of  the  line  from  (3,  2) 
to  the  middle  point  of  this  chord. 

12.  Find  the  tangents  to  the  ellipse  7  x'^  +  8y'^  =  oQ  which  make  the 
angle  tan~i3  with  the  line  x  ■}-  y  +  1  =0. 

13.  Find  the  product  of  the  two  segments  into  which  a  focal  chord  is 
divided  by  the  focus  of  an  ellipse. 

14.  Find  the  equation  of  a  tangent,  and  also  of  a  normal,  to  the  ellipse 
x^  -i-  4:  y^  =  IQ,  each  parallel  to  the  line  '6  x  —  4:  y  =  5. 

15.  Find  the  pole  of  the  line  3x  —  4:y  =  o  with  reference  to  the  ellipse 
a:^  +  4  ?/2  =  16 ;  also  the  intercepts  on  the  axes  made  by  a  line  through  the 
pole  and  perpendicular  to  the  polar. 

16.  Find  the  points  on  the  ellipse  h^x'^  +  a^y^  =  a%%  such  that  the  tan- 
gent makes  equal  (numerical)  angles  with  the  axes ;  such  that  the 
subtangent  equals  the  subnormal. 

146.  Auxiliary  circles.  Eccentric  angle.  The  circum- 
scribed and  inscribed  circles  for  the  ellipse  (Fig.  107)  are 
called  auxiliary  circles,  and  bear  an  important  part  in  the 
theory  of  the  ellipse.     Let  the  equation  of  the  ellipse  be 

-2+12=  1-  •••(!) 

The  circle  described  on  its  major  axis  as  diameter  is  called 
the  major  auxiliary  circle ;  its  equation  is 

2^4-^2  =  ^2.         .  .  .  (2) 

and  the  circle  on  the  minor  axis  as  diameter  is  the  minor 
auxiliary  circle  ;  its  equation  is 


145-146.] 


THE  ELLIPSE 


243 


If  ZAOQ  is  any  angle  (/>  at  the  center  of  the  ellipse,  with 
the  initial  side  on  the  major  axis,  and  the  terminal  side  cut- 
ting the  auxiliary  circles  in  R  and   §,  respectively  ;  and  if 




Y 

^^ 

y. 

B 

/ 

/ 

\ 

/^^ 

<:;:^ 

-^s 

p  ^ 

\ 

\ 

a: 

f 

/ 

\ 

\ 

\ 

0 

M' 
J 

JM 

j 

X 

tz 

5^ 

^ 

/ 

y 

/ 

1 

Fig. lOr. 

P  is  the  intersection  of  the  abscissa  LM  with  the  ordinate 
MQ,  then  P  is  a  point  on  the  ellipse. 

For  the  coordinates  of  P  are 

0M=  OQ  cos  (f>  and  MP  =  M'B  =  OR  sin  0, 
^.e.,  X  =  a  cos  ^,  ^  =  ^  sin  (f).       .       .       .       [60] 

Now  these  values  satisfy  the  equation  of  the  ellipse  ;  for, 
substituting  them  in  equation  (1),  gives 

a^  cos^  <f)  ,  h^  sin^  </>  9  •    ,     •   9  •       t 
^  H 7^  =  cos2  (^  -f  sm2  <^  =  1 ; 

hence  P  is  a  point  of  the  ellipse. 

The  points  P,  §,  and  i?  are  called  corresponding  points. 
The  angle  <^  is  the  eccentric  angle  of  the  point  P;*  and  the 

*  The  eccentric  angle  of  any  given  point  P  on  an  ellipse  is  readily  con- 
structed thus  :  produce  the  ordinate  3IP  to  meet  the  major  auxiliary  circle  in 
Q ;  the  angle  -40^  is  the  eccentric  angle  of  the  point  P. 


244  ANALYTIC  GEOMETRY  [Ch.  X. 

two  equations  [60]  are  equations  of  the  ellipse  in  terms  of 

the  eccentric  angle,  for  together  they  express  the  condition 

that  the  point  F  is  on  the  ellipse  (1).* 

Since,  in  the   figure,  A  OM' R   and   OMQ  are    similar,  it 

follows  that 

MP:MQ=  OB:  OQ  =  b:a, 

and  OM':  0M=  OB:  OQ  =  h:a; 

that  is,  the  ordinate  of  any  point  on  the  ellipse  is  to  the  ordi- 
nate of  the  corresponding  p>oint  on  the  major  auxiliary  circle  in 
the  ratio  (h  :  a)  of  the  semi-axes.  Similarly  for  the  abscissas 
of  the  corresponding  points  B  and  P. 

147.   The  subtangent  and  subnormal.     Construction  of  tan- 
gent and  normal. 

Let  -2  +  7-2  =  1         .  .  .  (1) 

a'^      0^ 

be  a  given  ellipse, 

then  ^  +  1^  =  1,         ...  (2) 

a^        0^ 

is  the  tangent  to  it  at  a  point  P-^=(a^j,  y^.  Let  this  tangent 
cut  the  rr-axis  at  the  point  T.     Draw  the  ordinate  MPy 

Then  the  subtangent  is,  by  definition,  TM\  and  its  numer- 
ical value  is 

MT=OT- OMi 

but,  from  equation  (2),  0T=  —  \  and  OM==x^\ 

a^ 
hence  MT= x 


tAy-t 


i.e.,  TM='^ 


x^  —  c^ 


*  The  equations  [60]  are  of  great  service  in  studying  the  ellipse  by  the 
methods  of  the  differential  calculus. 


146-147.] 


THE   ELLIPSE 


245 


Hence  the  value  of  the  subtangent,  corresponding  to  any 
point  of  the  ellipse  whose  equation  is  (1),  depends  only  upon 
the  major  axis,  and  the  abscissa  of  the  point ;  therefore,  if  a 
series  of  ellipses  have  the  same  major  axis^  tangents  drawn  to 
them  at  the  points  having  a  common  abscissa  will  cut  the  major 
axis  (extended^  in  a  common  point. 


Fig. 108. 


This  fact  suggests  a  method  for  constructing  a  tangent 
and  normal  to  an  ellipse,  at  a  given  point :  draw  the  major 
auxiliary  circle ;  at  Q  on  this  circle,  and  in  MP^  extended, 
draw  a  tangent  to  the  circle.  This  will  cut  the  axis  in  T  \ 
and  Pj Twill  be  the  required  tangent  of  the  ellipse  at  P^. 
The  normal  P^iV^may  then  be  drawn  perpendicular  to  P^T. 

The  equation  of  the  normal  through  P^  is  (cf.  eq.  [51]) 

OjII 

y  ~  y\  —  p^  (^  —  ^i) ; 

therefore  the  a;-intercept  of  the  normal  at  that  point  is 


ON^ 


a'^-W' 


—x-i  =  e^x 


a' 


v 


246  ANALYTIC  GEOMETRY 

But  the  subnormal  corresponding  to  F^  is 
M]V=  ON-  OM, 
and  0M=  x^ ; 

therefore 


[Ch.  X. 


MN=  "^-^x.  -  X. 


a'- 


= x^  =  (e^  —  l)a7j. 

Note.  From  the  value  of  ON  it  follows  that  the  normal  to  an  ellipse 
does  not,  m  general,  pass  through  the  center,  but  passes  between  the 
center  and  the  foot  of  the  ordinate ;  the  extremities  of  the  axes  of  the 
curve  being  exceptional  points.  If,  however,  a  =  b,  then  e  =  0,  the  curve 
is  a  circle,  and  every  normal  passes  through  the  center  (cf.  Art.  85). 

148.  The  tangent  and  normal  bisect  externally  and  inter- 
nally, respectively,  the  angles  between  the  focal  radii  of  the 
point  of  contact. 


,EiG.109. 


f 


Let  the  equation  of  the  given  ellipse  be  -^  +  t2  =  ^  J  ^^^^ 
let  -Fj  and  F^  be  the  foci,  and  F^  =  (a;^,  y^)  any  given  point 
on  the  curve.  Draw  the  tangent  TF^,  the  normal  P^iV,  and 
also  the  lines  F^F^  and  F^F^  W. 


147-149.]  THE  ELLIPSE  247 

Then  F^N  =F^0-\-  0N=  ae  +  A^  [Art.  147] 

=  e(a  +  ea^j), 

NF^   =  OF^^  -  0N=  ae  -  e\ 
=  e(a  —  ex^^  , 

also  F^F^  =  a  +  ex^,  [Art.  144] 

and  -^2^1  =  a  —  exy 

Hence  F^JST :  iVT2  =  ^^P^ :  F^F^ ; 

and,  by  a  theorem  of  plane  geometry,  this  proportion  proves 
that  the  normal  F^JY  bisects  the  angle  F^F^F^  between  the 
focal  radii.  Again,  since  the  tangent  is  perpendicular  to 
the  normal,  the  tangent  F^T  will  bisect  the  external  angle 
FoFiW. 

This  proposition  leads  to  a  second  method  of  constructing 
the  tangent  and  normal  to  an  ellipse  at  a  given  point 
(cf.  Art.  147).  First  determine  the  foci,  F^  and  F^  (Art. 
144),  then  draw  the  focal  radii  to  the  given  point  and 
bisect  the  angle  thus  formed,  —  internally  for  the  normal, 
externally  for  the  tangent. 

149.    The  intersection  of  the  tangents  at  the  extremity  of  a  focal  chord. 

If  P'  =  (x',  y')  be  the  intersection  of  two  tangents  to  the  ellipse 

a2  "*■  62  -  -^' 
the  equation  of  their  chord  of  contact  is  (Art.  126) 

x'x     y'y 


h^ 


=  1 (1) 


If  this  chord  passes  through  the  focus  Fo^  (^^5  0),  its  equation  must 
be  satisfied  by  the  coordinates  of  F^]  therefore 

x'ae  _  ^      .      ^     ^ 
-^  =  1,    i.e.,x'  =  -, 


248  ANALYTIC  GEOMETRY  [Ch.  X. 

and  the  point  of  intersection  P'  is  on  the  line,  a:=  -;  i.e.,  on  the  directrix 

corresponding  to  the  focus  F^.     Similarly,  if  the  chord  passes  through 

the  focus  i^"j  =  (—  ae,  U),  the  poiut  P'  is  on  the  directrix  x  = 

Hence,  the  tangents  at  the  extremities  of  a  focal  chord  intersect  upon  the 
corresponding  directrix. 

Again,  the  line  joining  the  intersection  P'  =  (  -,  y' )  to  the  focus  has 

the  slope 

y^-  y\_    y'    _     ^y'     _  ^^y' . 


m 


a  a  (1-6-2)        62  ' 

ae        ^  ^ 


e 

while  the  slope  of  the  focal  chord  (1)  is 

&V  _       62  ^ 
a^y'  aey' ' 

hence  ra' = -\ 

m 

and  therefore  the  line  joining  the  focus  to  the  intersection  of  the  tangents  at 
the  ends  of  a  focal  chord  is  perpendicular  to  that  chord. 

150.  The  locus  of  the  foot  of  the  perpendicular  from  a  focus  upon  a 
tangent  to  an  ellipse.  Let  the  equation  of  a  tangent  to  the  ellipse 
(Art.  143),  whose  equation  is 

^.+1  =  1.       .        .        .        (1) 


be  written  in  the  form        y  =  mx  -\-  y/ahn^  +  h\  .         .  .  (2) 

Then  the  equation  of  a  perpendicular  to  (2),  through  the  focus  {ae,  0),  is 

y  = (a:  — ae),      i.e.,  x  +  my  =  ae.      .      .       .       (3) 

If  P'=(x',  y')  is  the  point  of  intersection  of  (2)  and  (3),  it  is  re- 
quired to  find  the  locus  of  P' ;  i.e.,  to  find  an  equation  which  will  be 
satisfied  by  the  coordinates  x',  y',  whatever  the  value  of  m;  this  must 
be  an  equation  involving  x'  and  y',  but  free  from  m.  Since  P'  is  on 
both  lines  (2)  and  (3), 


therefore  y'  —  mx'  =  Va^m'^  +  b%  ...  (4) 

and  x'  +  my'  =  ae.  .  .  -  (5) 


149-151.]  THE  ELLIPSE  2J:9 

The  elimination  of  m  is  accomplished  most  easily  by  squaring  each 
member  of  equations  (4)  and  (5),  and  adding: 

this  gives  (1  +  in^)  j;'-  +  (1  +  w^)  y"^  =  ahn^  +  a^e^  +  h% 

i.e.,  (1  +  m2)(x'^  +/2)  =  «2  (^„^2  ^  1)^ 

whence,  x'^  +  jy'^  =  a-. 

Hence,  the  point  P'  is  on  the  circle 

a;2  +  ^2  _  ^2. 

that  is,  ^Ae  /ocus  of  the  foot  of  a  perpendicular  from  either  focus  upon  a  lau' 
gent  to  the  ellipse  is  the  major  auxiliary  circle. 

151.    The  locus  of  the  intersection  of  two  perpendicular  tangents  to  the 
ellipse. 

Let  the  equation  of  any  tangent  to  the  ellipse  —^-\-^—^  be  written 

in  the  form  (Art.  143)  

y  —  mx  =  'Va'^m^  +  b%  .  .  .  (1) 

then  the  equation  of  a  perpendicular  tangent  is 


m  m-^ 


i.e.,  my  +  x  =  Va^  +  6%w'^.  .  .  .  (2) 

Letting  P'  =  (x',  y')  be  the  point  of  intersection  of  these  two  tangents, 
(1)  and  (2),  it  is  required  to  find  the  locus  of  P'  as  m  varies  in  value; 
that  is,  to  find  an  equation  between  x'  and  y'  which  does  not  involve  m. 

Proceeding  as  in  Art.  150 ;  since  P'  is  on  both  lines  (1)  and  (2), 

therefore  y'  —  mx'  =  Va^m^  +  b% 


and  my'  +  x'  =  Va^  +  b'^m^. 

To  eliminate  m,  square  both  equations,  and  add :  this  gives 

(W2  +  1)  y2  +  (^^2  ^  1)  ^./2  ^  Qn2  +  1)  a2  +  ^„^2  +  J)  ^2^ 

i.e.,  x'^  +  y'^  =  a'^  -\-  b^. 

Therefore,  the  point  of  intersection  of  perpendicular  tangents  is  on 
the  circle 

a;2  -f  y-2  =  rt2  +  J2^  ^  ^  ^  [-Q1J 

which  is  called  the  director  circle  for  the  ellipse.     The  locim  of  the  inter- 
section of  two  perpendicular  tanyents  to  an  ellipse  is,  then,  its  director  circle. 


250  ANALYTIC  GEOMETRY  [Ch.  X. 

EXERCISES 

1.  Prove  that  the  two  tangents  drawn  to  an  ellipse  from  any  external 
point  subtend  equal  angles  at  the  focus. 

2.  Each  of  the  two  tangents  drawn  to  the  ellipse  from  a  point  on  the 
directrix  subtends  a  right  angle  at  the  focus. 

3.  A  focal  chord  is  perpendicular  to  the  line  joining  its  pole  to  the 
focus.     Show  that  this  is  also  true  for  a  parabola. 

4.  The  rectangle  formed  by  the  perpendiculars  from  the  foci  upon  any 
tangent  is  constant ;  it  is  equal  to  the  square  of  the  semi-minor-axis. 

5.  The  circle  on  any  focal  distance  as  diameter  touches  the  major 
auxiliary  circle. 

6.  The  perpendicular  from  the  focus  upon  any  tangent,  and  the  line 
joining  the  center  to  the  point  of  contact,  meet  upon  the  directrix. 

7.  The  perpendicular  from  either  focus,  upon  the  tangent  at  any  point 
of  the  major  auxiliary  circle,  equals  the  distance  of  the  corresponding 
point  of  the  ellipse  from  that  focus. 

8.  The  latus  rectum  is  a  third  proportional  to  the  major  and  minor 
axes, 

9.  The  area  of  the  ellipse  is  trah. 

Suggestion.  Employ  the  fact,  proved  in  Art.  146,  that  the  ordinate 
of  an  ellipse  is  to  the  corresponding  ordinate  of  the  major  auxiliary 
circle  as  h :  a,  and  thus  compare  the  area  of  the  ellipse  with  that  of  its 
major  auxiliary  circle. 

152.  Diameters.  As  already  shown  in  Articles  129  and 
139,  the  definition  of  a  diameter  as  the  locus  of  the  middle 
points  of  a  system  of  parallel  chords  leads  directly  to  its 
equation. 

Let  m  be  the  slope  of  the  given  system  of  parallel  chords 
of  the  ellipse  whose  equation  is 

$+$=h  .    (1) 

and  let  2/  =  mx  +  c  .         ,         .       (2) 


151-152.] 


THE  ELLIPSE 


251 


be  the  equation  of  one  of  these  cliords,  which  meets  the  curve 
in  the  two  points  P^  =  (x^,  y{)  and  P2=(%  2/2)-  -^^^ 
P'  =  (a;',  ^'),  be  the  middle  point  of  this  chord,  so  that 


,  _  ^1  +  x^      ,  _  yx^y% 

^  -       2     '     ^  ~       2      • 


.      (3) 


/ 

r 

I 

/ 

B    yp// 

/// 

t/^ 

/ 

//// 

y/ 

/A 

^ 

<^ 

m 

/  \ 

\ 

\a 

X 

-/ 

^\ 

^ 
z 

/  / 

V^ 

y 

1 

'  W 

/ 

B' 

Fig 

.no 

The  coordinates  of  P^  and  P^  ^^®  found  by  solving  (1) 
and  (2)  as  simultaneous  equations,  therefore  the  abscissas 
a^j  and  x^  are  the  roots  of  the  equation 

{c^rr^  +  5^)  a;^  +  2  aP'cmx  +  a^c^  —  ^252  _  q^    ^   ^ 
and  the  ordinates  y-^  and  y^  are  roots  of  the  equation 

Hence,  by  Art.  11,  the  coordinates  of  P'  are 

a^rn^  _!_  52'     i'       a2^j^2  _|_  52*  • 

Now,  by  varying  the  value  of  c,  equation  (6)  gives  the 
coordinates  of  the  middle  point  for  each  of  the  chords  of  the 
given  set.  It  is  required  to  find  the  locus  of  P'  for  all 
values  of  <?,  i.e.,  to  find  an  equation  satisfied  by  x'  and  y'. 


(4) 


(5) 


(6) 


252  ANALYTIC  GEOMETRY  [Ch.  X. 

and  not  dependent  upon  the  value  of  c.     If  x'  be  divided  by 
^',  the  c  is  eliminated  from  the  equations  (6),  giving 

y  =  -j2'»-        ...         (7) 

Therefore  the  coordinates  of  the  middle  point  of  every 
chord  of  slope  m  satisfy  the  equation 

X  a^ 

y        i>^ 

or,  y=-—-x\  .         .  .        [62] 

which   is   therefore  the  equation  of  the  diameter  bisecting 
the  chords  of  slope  m. 

The  form  of  equation  [62]  shows  that  every  diameter  of 
the  ellipse  passes  through  the  center, 

153.  Conjugate  diameters.  Since  every  diameter  passes 
through  the  center  of  the  ellipse,  and  since,  by  varying  tlie 
slope  m  of  the  given  set  of  parallel  chords,  the  corresponding 
diameter  may  be  made  to  have  any  required  slope,  therefore 
it  follows  that  every  chord  which  passes  through  the  center  of 
an  ellipse  is  a  diameter^  corresponding  to  some  set  of  parallel 
chords.  In  particular,  that  one  of  the  set  of  chords  given 
by  equation  (2),  Art.  152,  which  passes  through  the  center, 
—  ^^g.,  the  chord  whose  equation  is 

y  =  mx,  ...  [63] 

is  a  diameter.     This  diameter  bisects  the  chords  parallel  to 
the  line  [62];  for  if  m'  be  the  slope  of  the  line  [62], 

then  m'  = — , 

a- m 

hence,  nm'  = ;       •         •         •         [64] 

a^ 


152-154.]  THE  ELLIPSE  253 

and  this  equation  expresses  the  condition  that  line  [62], 
which  has  the  slope  m\  shall  bisect  the  chords  of  slope  m 
(Art.  152).  But  conversely,  it  expresses  also  the  condition 
that  the  line  [63]  which  has  the  slope  m  shall  bisect  the 
chords  of  slope  wJ .  Hence  each  of  the  lines  [62]  and  [63] 
bisects  the  chords  parallel  to  the  other.  Hence,  if  one 
diameter  bisects  the  chords  parallel  to  a  second^  then  also  the 
second  diameter  bisects  the  chords  parallel  to  the  first.  Such 
diameters  are  called  conjugate  to  each  other. 

Each  line  of  the  set  of  parallel  chords  in  general  cuts  the 
ellipse  in  two  distinct  points,  and  the  further  the  chord  is 
from  the  center,  the  nearer  these  two  points  are  to  each 
other,  and  to  their  mid-point.  In  the  limiting  position,  the 
chord  becomes  a  tangent,  with  the  two  intersection  points 
and  their  mid-point  coincident  at  the  point  of  tangency. 
Therefore,  the  tangent  at  the  end  of  a  diameter  is  parallel  to 
the  conjugate  diameter.  This  property,  with  that  of  Art.  152, 
suggests  a  method  for  constructing  conjugate  diameters : 
first  draw  a  tangent  at  an  extremity  of  a  given  diameter 
(Art.  147),  then  a  line  drawn  parallel  to  this  tangent  through 
the  center  of  the  ellipse  is  the  required  conjugate  diameter. 
(See  Fig.  111.) 

154.  Given  an  extremity  of  a  diameter,  to  find  the  extremity  of  its 
conjugate  diameter. 

Let  P^=(Xp?/j)  be  an  extremity  of  a  given  diameter  (Fig.  ill),  then 
P^=(jx^,  -y^  will  be  the  other  extremity.  Let  P/  =  (a:/,  3//)  and 
P^  =  (jx^,  -y{)  be  the  extremities  of  the  conjugate  diameter.  Let  the 
equation  of  the  ellipse  be 

then  the  equation  of  the  given  diameter  P^Pg  i^ 

y=|^,  .  »  «  (2) 


254 


ANALYTIC  GEOMETRY 


[Ch.  X. 


and  that  of  the  conjugate  diameter  Pi'P^j  through  the  center  and  parallel 
to  the  tangent  at  Pj  is 


1 

7 

'^-^ 

^^9. 

* 

r 

<^ 

TNT 
/  /  ' 
/  /   \ 

V 

/ 

^^jif, 

^^^-- — 

The  coordinates  of  P/  and  P{,  in  terms  of  Xj,  ?/i,  a,  and  &,  are  given  by- 
equations  (1)  and  (3),  considered  as  simultaneous ;  hence,  eliminating 
y  between  these  equations,  and  remembering  that  the  point  P^  is  on  the 
ellipse  (1)  and  that  therefore  If-x^  +  d-if'  =  a^P'^  the  abscissas  of  the 
points  Pj'  and  Pg'  are  given  by  the  equation 


,2  -t]il> 


x"  = 


&2 


I.e., 


x{  =  -p^    and   x^=-^yy 


Substituting  these  values  in  equation  (3),  gives  for  the  corresponding 
ordinates, 

y{=-Xx     and     y{  =  --^r 
Therefore  the  required  extremities  of  the  conjugate  diameter  are 

155.   Properties    of    conjugate    diameters    of    the    ellipse. 

(a)  It  has  been  seen   (Art.   153)  that  two   diameters  are 
conjugate  when  their  slopes  satisfy  the  relation 


mm  = -. 

a"" 


(1) 


154-155.]  THE  ELLIPSE  255 

It  follows,  since  the  product  of  their  slopes  is  negative, 
that  with  the  exception  of  the  case  where  one  diameter  is 
the  minor  axis  itself,  conjugate  diameters  do  not  both  lie  in 
the  same  quadrant  formed  by  the  axes  of  the  curve. 

(/S)  From  the  definition  (Art.  153)  it  is  evident  that  the 
minor  and  major  axes  of  the  ellipse  are  a  pair  of  conjugate 
diameters,  and  they  are  at  right  angles  to  each  other.  Per- 
pendicular lines,  however,  in  general,  fulfill  the  condition 

mm'  =  —  1 ;         .         .         .         (2) 

hence,  in  general,  equation  (2)  is  not  consistent  with  equa- 
tion (1)  for  other  values  of  m  and  m'  than  0  and  oo,  —  the 
slopes  for  the  axes  of  the  curves.  But  for  b^  —  ol^^  ^.e.,  for 
the  circle,  it  is  clear  that  every  pair  of  conjugate  diameters 
satisfy  equation  (2),  and  are  therefore  perpendicular  to  each 
other.  Hence,  tlie  major  and  minor  axes  of  the  ellipse  are 
the  only  pair  of  conjugate  diameters  that  are  perpendicular  to 
each  other. 

(7)  If,  in  Fig.  Ill,  the  lengths  of  the  conjugate  semi-axes 
be  a^  =  CPy,  b'  =  0P^\  then,  since 

b^x-^  4-  a^y^^  =  a%'^^  a''^  =  xf  +  y^., 

and  6'2  =  ^V^; 

0^  a'^ 

therefore        a'^  +  V^  =  ^'^i' +/lf^'  +  oV+^V 

=  a^  +  b^',  .  .  .  (3) 

i.e.,  the  sum  of  the  squares  of  ttvo  conjugate  semi-diameters  is 
constant ;  it  is  equal  to  the  sum  of  the  squares  of  the  two  semi- 
axes. 


256  ANAL  r TIC  GEOMETBY  [Ch.  X. 

(3)  Referring  again  to  Fig.  Ill,  where  C^iV  is  perpen- 
dicular to  the  tangent  at  Pj,  the  conjugate  diameters  .^1^2 
and  P^^^  intersect  at  an  angle  i/r  such  that 

-f  =Z  P^CP{  =  90°  +Z  P^CN% 

sin  A/r  =  cos  Z  P^CN=  ^' 

^P\ 

But,  by  Art.  64,  since  the  equation  of  the  tangent  at 
P^  is  IP'x^  +  (^y\y  —  oP-h^^ 

aH^  ah  ah 


CN  = 


V6V  +  «Vi^       ^\aH.^      b%^       ^ 


f ' 


^    P    ^    a 

but  OP^  =  a', 

ah 


hence  sini/rzz:-— ,  .  .  .  (4) 


and  the  angle  hetween  tivo  conjugate  diameters  is  sin~^——. 

ah' 

(e)  Tangents  at  the  extremities  of  a  pair  of  conjugate 
diameters  form  a  parallelogram  circumscribed  about  the 
ellipse  ;  its  sides  are  parallel  to,  and  equal  in  length  to, 
the  conjugate  diameters.  Since  the  area  of  a  parallelogram 
is  equal  to  the  product  of  its  adjacent  sides  and  the  sine  of 
the  included  angle,  therefore  the  area  of  this  circumscribed 
parallelogram  is  4:a'h'  sin-v^,  which,  by  (4),  equals  4  ah. 

That  is,  the  area  of  the  parallelogram  constructed  upon  any 
two  conjugate  diameters  is  constant;  it  is  equal  to  the  area  of 
the  rectangle  upon  the  axes. 

(f)  A  simple  relation  exists  between  the  eccentric  angles 
of  the  extremities  of  two  conjugate  diameters. 

Let  the  eccentric  angle  of  Pi=  (x-^.y-^  be  (/>!  (Fig.  112), 
and  of  P2  =  (^2^  ^2)  ^®  <^2  j  then  the  slopes  of  the  conjugate 
diameters  may  be  written  (cf.  Art.  146), 


155-156.] 
for  (7Pi, 
and  for  (7^2? 
But 


THE  ELLIPSE 

Vi       h  sin  <bi 
Xi      a  cos  <pi 

x^     a  cos  <^2 


257 


mm' =  — 


a 


2' 


[Art.  155  (a)] 


Y    v^ 

%'''^ 

N 

/'J^K^"^ 

•C  ^^'^N 

4 

i^i^^^.'- 

sx/^ 

\"^^->. 

4 

'         ^^ 

^K^i'?', 

\a     ^~^'^ 

^7     X 

1 

ilfj                   y^ 

\ 

^S^^ 

\ 

^    ~- 

— -'''    Fig 

US. 

hence 


giving 


W"  sin  (^1  sin  (f)2  _  _^^ 

a^  cos  (^1  cos  (^2  ^^ 

sin_^i_sin_^  _  _  -j  ^ 

cos  (^1  COS  <^2 


that  is,  sin  ^2  sin  (f)i  +  cos  (/)2  cos  ^^  =  0, 

whence  cos((/)2  —  (f>i)=  0. 

Therefore  <^2  -  01  =  90°, 

and  ^Ae   eccentric  angles  of  the   extremities  of  two  conjugate 
diameters  differ  by  a  right  angle. 

156.  Equi-conjugate  diameters.  If  two  conjugate  diameters  be  equal 
to  each  other,  e.g.,  if  CP^  —  CP^  (see  Fig.  112),  then  the  properties  given 
in  the  preceding  article  lead  to  other  simple  onesv 

Let  0j  be  the  eccentric  angle  of  Pj,  then  <^j  +  90°  is  the  eccentric  angle 
for  Pgi  hence  the  coordinates  of  P^  and  Pg  ^^^  (^  ^o^  ^v  ^  ^^^  4*i)  ^^^ 
(-a  sin  (ji^,  b  cos  0^),  and  since 

a'=b', 

TAN.    A>r.    GEOM.  —  17 


258  ANALYTIC  GEOMETRY  [Ch.  X. 

therefore         a^  cos^  <f>^  +  b"^  sin^  <^^  =  a^  sin2  <^^  +  fts  qqq2  ^^^ 
i.e.,  taii2  <^i  =  1. 

Hence  <^i  =  45°  or  135° 

for  the  extremities  of  equi-con jugate  diameters,  and  the  extremities  are 

p  _/       &      \      p  =/         ^     \ 
\       a     J  \         a     / 

The  equations  of  these  diameters  are 

b           .             b 
y  =  -  X,  and  y  = x. 

a  a 

Evidently  these  lines  are  the  diagonals  of  the  rectangle  formed  on  the 
axes  of  the  curve. 

By  Art.  155,  (y),  the  length  of  each  equi-conjugate  semi-diameter  is 


2  +  62 


EXERCISES 

1.  Find  the  diameter  of  the   ellipse   —  +  ^-  =  1  which  bisects  the 

^       16       9 

chords  parallel  to  the  line  dx  +  5y-\-7  =  0. 

2.  Find  the  diameter  conjugate  to  that  of  exercise  1. 

3.  Show  that  the  lines  2 x  —  y  =  0,  x  +  dy  =  0  are  conjugate  diame- 
ters of  the  ellipse  2  x'^  -\-  3  y^  =  4:. 

4.  For  the  ellipse  b^x^  +  a^y^  =  a%\  write  the  equations  of  diameters 

conjugate  to  the  line 

(a)  ax  =  by,     (/?)  bx  =  ay. 

5.  Prove  that  the  angle  between  two  conjugate  diameters  is  a 
maximum  when  they  are  equal. 

6.  Show  that  the  pair  of  diameters  drawn  parallel  to  the  chords  join- 
ing the  extremities  of  the  axes  are  equal  and  conjugate. 

7.  What  are  the  equations  of  the  pair  of  equi-conjugate  diameters 
of  the  ellipse  lQy^  +  9x^  =  144? 

8.  Two   conjugate   diameters   of   the  ellipse h  ^  =  1    have    the 

slopes  I  and  —  f,  respectively;  find  their  lengths. 

9.  Given  the  ellipse  x'^-^5y^=5,  find  the  eccentric  angle  for  the 
point  whose  abscissa  is  1.  Also  find  the  diameter  conjugate  to  the  one 
passing  through  this  point. 


156-157.]  THE  ELLIPSE  259 

10.  Given  the  ellipse  3  a;2  +  4  ?/2  =  12,  find  the  conjugate  diameters 
for  the  point  whose  eccentric  angle  is  30'"\ 

11.  Find  the  lengths  of  the  diameters  in  exercise  10. 

12.  The  lengths  of  the  chord  joining  the  extremities  of  any  two  con- 
jugate diameters  of  the  ellipse  ^  +  ^  =  1  is  Va2+  62  +  a%2sin2<^. 
Find  its  greatest  value.     What  is  the  corresponding  value  of  <^  ? 

13.  The  area  of  a  triangle  inscribed  in  an  ellipse,  if  </)p  <^2»  ^z  ^^  *^® 
eccentric  angles  of  the  vertices,  is 

^ab  [sin  {4>^  -  cfi^)  +  sin  (<^3  -  c^J  +  sin  (</)i  -  (^2)]- 

14.  (xiven  the  point  (-3,  "1)  on  the  ellipse  x^  +  3  ?/2  =  12 ;  find  the 
corresponding  point  on  the  major  auxiliary  circle,  and  also  find  the 
eccentric  angle  of  the  given  point. 

15.  Find  the  polar  of  the  focus  of  an  ellipse  with  reference  to  each 
auxiliary  circle. 

16.  Find  the  pole  of  the  directrix  of  the  ellipse  with  reference  to  each 
auxiliary  circla. 

17.  Prove  analytically  that  tangentsat  the  ends  of  any  chord  intersect 
on  the  diameter  which  bisects  that  chord. 

157.  Supplemental  chords.  The  chords  drawn  from  any  point  of 
an  ellipse  to  the  extremities  of  a  diameter  are  called  supplemental  chords. 
Such  chords  are  always  parallel  to  a  pair  of  conjugate  diameters,  since 
their  slopes  satisfy  the  relation 

,62 

mm  = 

a2 

For  if  P^  =  (x^,  y^)  and  Po  =  (-^i,  -  y-^  be  the  extremities  of  a 
diameter,  and  P'  ={x',  y')  be  any  other  point  of  the  ellipse,  and  m  and 
m'  the  slopes  of  the  chords  P'P^  and  P'Po'  respectively, 

then  m=y'-y^-  ^'-y'-+Jh 

therefore 
But 

and 


1          '    " 

x'  +  x^ 

mm! 

y"-y,' 

~  x'^  -  x^ 

^      y  2 
a2      62 

=  1, 

^i\yx' 
a2+  62 

=  1; 

260  ANALYTIC  GEOMETBT  [Ch.  X 

hence,  by  subtraction, 

tnaij  IS,  /Q        9  —       o » 

'  x^  —  x-^^         or 

hence  mm'  = -• 

a2 

Therefore,  supplemental  chords  are  parallel  to  a  pair  of  conjugate 
diameters. 

For  the  special  case  when  a  =  b,  the  product  of  the  slopes  becomes 
mm'  =  —  1,  and  therefore  the  supplemental  chords  are  perpendicular ;  in 
other  words,  the  angle  inscribed  in  a  semicircle  is  a  right  angle. 

158.  Equation  of  the  ellipse  referred  to  a  pair  of  conjugate  diameters. 
In  the  simplest  form  for  the  equation  of  the  ellipse,  viz., 

-'+'!^=i-     •      •      •     "(1) 

the  coordinate  axes  are  the  axes  of  the  curve.  These  axes  are  conjugate 
diameters,  and  they  are  the  only  pair  which  are  at  right  angles  to  each 
other  (cf.  Art.  155,  /3).  It  is  desired  now  to  find  the  equation  of  the 
curve  referred  to  any  pair  of  conjugate  diameters,  as  P2P1  and  P^P^,  in 
Fig.  111.  With  the  notation  of  Art.  154,  let  6  and  6'  be  the  angles  the 
new  X-axis,  CPj,  and  the  new  ?/-axis,  CP^,  make  with  the  old  x-axis,  re- 
spectively; they  satisfy  the  relation  [64], 

tan  ^  tan  ^' =  - -.         ...  (2) 

a^ 

The  lengths  of  the  conjugate  semi-diameters  are  a'  =  CP^  and 
b'  =  CP/. 

Then,  by  Art.  73,  the  equations  for  transformation  to  the  new  axes  are 

X  =  x'  cos  6  +  y'  cos  0',  y  =  x'  sinO  +  y'  sin  0',      ...     (3) 

and  after  transformation  equation  (1)  becomes 

/cos^  _j_  sin2^\  ^,2  _^  2  /cosOcosO'  _^  sin  ^  sin  ^^\^,, 
\    a2  b^    J  \        a^  b'        J 

+  (£2^H_^),3.i.      ...      (4) 

But,by(2),  sme^mO'  ^_b^^ 

cos  0  cos  6'         a^ 


157-159.]  THE  ELLIPSE  261 

,                                        sin  0  sin  0'  ,  cos  0  cos  6'      /x 
hence + =  0, 

and  equation  (4)  reduces  to 

{s9^+?i}^y+(i2^+^jfLy^=i. ...  (5) 

From  equation  (5)  it  is  seen  that  the  curve  is  obliquely  symmetrical 
with  respect  to  the  new  axes.  Moreover,  since  ±  a'  and  ±  h'  are  the 
intercepts  on  the  new  axes,  equation  (5)  may  be  further  simplified : 

for         -    (^-^+^y^=i^ 

and  f£24^  +  «i^V'^  =  l; 


&2 


cos2(9  ,  sin2^       1      cos2^'  ,  sin2^'       1 
hence  — —  +  ——-  =  — ?   — -—  +  =  rrr? 

and  equation  (5)  may  be  written 

^1  +  1^  =  1 [65] 

This  is  the  required  equation  of  the  ellipse  when  referred  to  any  pair  of 
conjugate  diameters.  It  is  evident  that  propositions  which  were  derived 
for  the  standard  form  (1)  without  reference  to  the  fact  that  the  axes 
were  rectangular,  hold  equally  for  equation  [65]  ;  e.g.,  the  equation  of  a 

tangent  at  the  point  (x^,  y{)  of  the  curve  is  ^  +  ^  =  1- 

Equation  [65]  states  a  geometric  property  of  the  ellipse  entirely 
analogous  to  that  of  Art.  112.  It  is  left  to  the  student  to  express  this 
property  in  words. 

If  the  ellipse  is  referred  to  equi-conjugate  diameters,  so  that  a'  =  &', 

its  equation  will  be 

x^  +  y^  =  a'^.  .  .  .  [6Q-] 

This  is  the  same  form  as  the  simplest  equation  of  the  circle,  but  here 
the  axes  are  oblique,  and  the  equation  represents,  not  a  circle,  but  an 
ellipse. 

159.   Ellipse  referred  to  conjugate  diameters;  second  method. 

If  the  ellipse 

^  +  ^  =  1  .  .  .  (1) 

is  transformed  to  a  paii*  of  conjugate  diameters,  its  equation  after  trans- 
formation (Art.  73)  must  be  of  the  form 

Ax^-{-2Hxy  +  By^  =  l.         ...         (2) 


262  ANALYTIC  GEOMETRY  [Ch.  X. 

But,  since  eacli  chord  parallel  to  either  axis  is  bisected  by  the  other, 

therefore,  if  (a:^,  y^)  is  a  point  on  the  curve,  then  (-x^,  +  y■^)  must  also 

be  on  the  curve ; 

i.e.,  if  Axj^  +  2  Hx^y^  +  By^^  =  1, 

then  Ax^^-2Hx^y^-\-By^^  =  l, 

and,  consequently,  H  =  0. 

Again,  (a',  0)  and  (0,  &')  are  points  on  the  curve ; 

hence  Aa'-^  =  1,        Bb''^  =  l; 

therefore,  equation  (2)  becomes 

rt'2         fe'2 

This  method  illustrates  how  analytic  reasoning  may  often  be  used 
to  shorten  or  perhaps  obviate  the  .  algebraic  reductions  involved  in  a 
proof.  With  the  similar  methods  of  Arts.  39  and  40,  it  will  suggest 
to  the  reader  the  power  and  interest  of  what  are  called  the  modern 
methods  in  analytic  geometry. 

EXAMPLES    ON    CHAPTER   X 

1.  Find  the  foci,  directrices,  eccentricity  of  the  ellipse  4  a:^  +  3  2/-  =  5. 

2.  Find  the  area  of  the  ellipse  4a;2  +  3^2  _  5  (^^f.  Art.  151,  Ex.  9). 

3.  Show  that  the  polar  of  a  point  on  a  diameter  is  parallel  to  the 
conjugate  diameter. 

4.  Find  the  equations  of  the  normals  at  the  ends  of  the  latus  rectum, 
and  prove  that  each  passes  through  the  end  of  a  minor  axis  if  e*  4.  g2_  \^ 

5.  Show  that  the  four  lines  from  the  foci  to  two  points  P^  and  P^ 
on  an  ellipse  all  touch  a  circle  whose  center  is  the  pole  of  ^1^2- 

6.  Tangents  are  drawn  from  the  point  (3,  2)  to  the  ellipse 

Find  the  equation  of  the  line  joining  (3,  2)  to  the  middle  point  of  the 
chord  of  contact. 

7.  Find  the  locus  of  the  center  of  a  circle  which  passes  through  the 
point  (0,  3)  and  touches  internally  the  circle  x^  +  ?/2  —  25. 

8.  Find  the  length  of  the  major  axis  of  an  ellipse  whose  minor  axis 
is  10,  and  whose  area  is  equal  to  that  of  a  circle  whose  radius  is  8. 


159.]  TRE  ELLIPSE  263 

9.  The  minor  axis  of  an  ellipse  is  6,  and  the  sum  of  the  focal  radii 
for  a  certain  point  on  the  curve  is  16;  find  its  major  axis,  distance 
between  foci,  and  area. 

10.  A  line  of  fixed  length  moves  so  that  its  ends  remain  in  the 
coordinate  axes;  find  the  locus  generated  by  any  point  of  the  line. 

11.  Find  the  locus  of  the  middle  points  of  chords  of  an  ellipse  drawn 
through  the  positive  end  of  the  minor  axis. 

12.  With  a  given  focus  and  directrix  a  series  of  ellipses  are  drawn ; 
show  that  the  locus  of  the  extremities  of  their  minor  axes  is  a  parabola. 

13.  Show  that  the  line  x  cos  a-\-  y  sin  a  =  j9  touches  the  ellipse 

e!  +  ^  =  i 

if  p^  =  a^  cos^  a  +  &2  sin^  a. 

14.  Find  the  locus  of  the  foot  of  the  perpendicular  drawn  from  the 

center  of  the  ellipse h  —  =  1  to  a  variable  tangent. 

15.  Prove,  analytically,  that  if  the  normals  to  an  ellipse  pass  through 
its  center,  the  ellipse  is  a  circle. 

16.  Find  the  locus  of  the  vertex  of  a  triangle  of  base  2  a,  and  such 
that  the  ]3roduct  of  the  tangents  of  the  angles  at  its  base  is  —  • 

17.  The  ratio  of  the  subnormals  for  corresponding  points  on  the 
ellipse  and  major  auxiliary  circle  is  — 

18.  Find  the  equation  of  the  ellipse  dx^  +  25y'^  =  225  when  referred 
to  its  equi-con jugate  diameters. 

19.  Normals  at  corresponding  points  on  the  ellipse,  and  on  the  major 
auxiliary  circle,  meet  on  the  circle  x'^  +  y^  =  (a  +  &)2. 

20.  Two  tangents  to  an  ellipse  are  perpendicular  to  each  other ;  find 
the  locus  of  the  middle  point  of  their  chord  of  contact. 

21.  If  Pj  is  a  point  on  the  director  circle,  the  product  of  the  distances 
of  the  center  and  the  pole,  respectively,  from  its  polar  with  respect  to 
the  ellipse  is  constant. 

The  tangents  drawn  from  the  point  P  to  an  ellipse  make  angles  0^ 
and  $2  with  the  major  axis ;  find  the  locus  of  P 

22.  when  0^  +  B^  =  2  a,  a,  constant. 

23.  when  tan  Oj  +  tan  ^g  =  c,  a  constant. 


264  ANALYTIC  GEOMETRY  [Ch.X.159. 

Find  the  locus  of  the  intersection  P  of  two  tangents 

24.  if  the  sum  of  the  eccentric  angles  of  their  points  of  contact  is  a 
constant,  equal  to  2  a. 

25.  if  the  difference  of  the  eccentric  angles  be  120°. 

26.  Find  the  locus  of  the  middle  points  of  chords  of  an  ellipse  which 
pass  through  a  given  point  Qi,  k). 

27.  Find  the  tangents  common  to  the  ellipse  ^  +  ^  =  1  and  its  mid- 

Q,       b 
circle  x^  +  y^  =:  ab. 


CHAPTER   XI 
The  Hyperbola,  ^-V^-i 

160.  Review.  The  definition  of  the  hyperbola  given  in 
Chapter  VIII  led  at  once  to  two  standard  forms  for  its  equa- 
tion, viz.  (of.  Arts.  116,  118): 

^2  _  ?/2  _  ^ 

when  the  axes  of  the  curve  are  coincident  with  the  coordi- 
nate axes  ;  and 

(jx  -  hf     (jy  -ky  ^. 

when  the  axes  of  the  curve  are  parallel  to  the  coordinate 
axes,  and  its  center  is  the  point  (A,  k^.  • 

A  brief  discussion  of  the  first  standard  form ^  =  1 

showed  that  the  curve  has  its  eccentricity  given  by  the  rela- 

^2  I  52 
tion   y^  =  a^(f-r),   i.e.,  by   e^  = ^— ;    its   foci  are  the 

two  points  (±(26,  0),  and  its  directrices  the  lines  x=±^ 

(Art.  116).  These  results  are  entirely  analogous  to  the 
corresponding  ones  for  the  ellipse,  if  it  be  remembered  that 
1  —  ^2  is  positive  for  the  ellipse,  while  e^  —  1  is  positive  for 
the  hyperbola. 

The  similarity  of  the  equations  of  the  hyperbola  and  the 
ellipse  leads  to  various  correspondences  in  the  analytic  prop- 
erties of  the  curves.     For  example,  the  equation 

265 


266  ANALYTIC  GEOMETRY  [Ch.  XI. 

represents  the  polar  of  the  point  (a;^,  ?/j)  Avith  respect  to  the 
hyperbola  ;  it  represents  the  chord  of  contact  if  the  point  is 
outside  the  hyperbola,  and  the  tangent  if  the  point  is  upon 
the  curve  (Arts.  126,  122).  Again,  by  the  method  sho^vn 
in  Art.  143,  merely  replacing  b'^  by  —  b^,  it  is  evident  that 


y  =  mx  ±  ^ahn^  —  b'^       .       .       .       [67] 

is  the  equation  of  a  tangent  to  the  hyperbola  in  terms  of  its 
slope  m.  The  student  will  be  able  in  like  manner  to  prove 
other  properties  of  the  hyperbola,  analogous  to  those  already 
shown  for  the  ellipse,  using  the  same  methods  of  derivation. 
It  was  shown,  however,  in  the  discussion  of  Chapter  VIII, 
as  also  in  Art.  48,  that  the  nature  of  the  hyperbola  appar- 
ently differs  widely  from  that  of  the  ellipse,  consisting,  as 
it  does,  of  two  open  infinite  branches  instead  of  one  closed 
oval.  It  is  desired  in  the  present  chapter  to  show  some  of 
the  most  important  properties  of  the  hyperbola  which  corre- 
spond to  similar  properties  in  the  ellipse  ;  and  also  to  prove 
some  special  properties  which  are  peculiar  to  the  hyperbola. 
For  the  most  part,  these  will  be  derived  for  the  hyperbola 

—  —  ^  =  1 ;  and  the  facts  summarized  above  will  be  assumed. 
a^      b^ 

161.  The  difference  between  the  focal  distances  of  any  point 
on  an  hyperbola  is  constant ;  it  is  equal  to  the  transverse  axis. 

The   hyperbola  —  —  ^  =  1    has   its    foci   at    the    points 
a^       b^ 

F^  =  (-ae,  0),  F^  =  (ae,  0),  with  b^  =  aV  _  ^2, 

Let  Pj  =  (2:^,  ?/-^)  be  any  given  point  on  the  curve,  so  that 

..  2  _  ^^^1   -l^ 


160-162.]  THE  HYPERBOLA  267 


Then  F^P{  =  (x^  +  ae^  +  ^^2  =  x^^  +  2  aex^  +  ah'^  +  ^^2 

=  ^2^2  4-  2  aea?!  H ^ — x.^  —  62 

=  a2g2  _|_  2  <35ea7j  +  62a;j2  _|_  ^2  _  ^2^2 

=  62^:^2  _^  2  aex-^  +  ^2, 
^.^.,  J^jPj  =  eo^i  + «.        .  .  .  (1) 

Similarly,  -^2-^1  =  ex^  —  a.       .         .         .         (2) 

These  expressions  for  the  focal  distances  of  a  point  on  the 
hyperbola  are  of  the  same  form  as  those  for  the  ellipse 
(Art.  144);  here,  however,  e>l. 

Subtracting  equation  (2)  from  equation  (1)  gives 

hence,  the  difference  hetiveen  the  focal  distafices  of  any  point 
on  an  hyperbola  is  constant;  it  is  equal  to  the  transverse  axis. 
If  the  foci  are  not  given,  they  may  be  constructed  as 
follows,  provided  the  semi-axes  of  the  curve  are  known  :  plot 
the  points  A=(a,  0)  and  ^  =  (0,  6);  then  with  the  center, 
of  the  hyperbola  as  center,  and  the  distance  AB  as  radius, 
describe  a  circle  ;  it  will  cut  the  transverse  axis  in  the 
required  foci  F^  and  -Fg,  for 


OF=AB  =  Va2  +  &2  =  Va2e2  =  ± 


ae. 


162.  Construction  of  the  hyperbola.  The  property  of  the 
preceding  article  might  be  taken  as  a  new  dehnition  of  the 
hyperbola,  viz.  :  the  hyperbola  is  the  locus  of  a  point  the  dif- 
ference of  whose  distances  from  two  fixed  poifits  is  constant. 
This  definition  leads  at  once  to  the  equation  of  the  curve 
(cf.  Ex.  6,  p.  67),  and  also  to  a  method  for  its  construc- 
tion. 


268  ANALYTIC   GEOMETRY  [Ch.  XI. 

(a)  Construction  hy  separate  points.  Let  A' A  be  the  given 
difference  of  the  focal  distances,  —  2. e.,  the  transverse  axis 
of  the  hyperbola, — and  F^  and  F^  the  given  fixed  ^Doints, 

the  foci.     With   either 

"-/>,         ,'-''V"""  focus,  say  j^i,  as  a  center, 

j!        A       R  and  a  radius  ^'i^>^'^, 

F,  II     describe   an   arc ;    then 

Fig.  113.  --V--'  ■-/--'  with  the  other  focus  as 

a  center,  and  a  radius 
AH  describe  an  arc  cutting  the  first  arcs  in  the  two  points 
Pi.  These  are  points  of  the  hyperbola.  Similarly,  as  many 
points  as  desired  may  be  obtained  and  then  connected  by  a 
smooth  curve, — approximately  an  hyperbola. 

(/S)  Construction  hy  a  continuously  moving  point ;  the  foci 
being  given.  Pivot  a  straight  edge  LM  at  one  focus  P^,  so 
that  FtMis  greater  than  the  trans-  ^ 

verse  axis  2  a  ;  at  iHf  and  the  other  ^^-<^^r^ 

focus  F^  fasten  the  ends  of  a  string  ..^l^^^^^^/l 

of  length  I,  such  that  F^M=l^2 a  ;    ^^:^^^^^     AJ   I 
then  a  pencil  P   held  against   the -^    ^         ^    ,,,    ^2 

^  °  _  FiG.lU. 

string  and  straight  edge  (see  Fig. 

114),  so  as  to  keep  the  string  always  taut,  will,  while  the 
straight  edge  revolves  about  F^,  trace  one  branch  of  the 
hyperbola.  By  fastening  the  string  at  the  first  focus  and 
the  straiglit  edge  at  the  second,  the  other  branch  of  the  curve 
can  be  traced. 

163.  The  tangent  and  normal  bisect  internally  and  exter- 
nally the  angles  between  the  focal  radii  of  the  point  of  contact. 

Let  Fy  and  F2  be   the   foci  of   the  hyperbola  —  — ^  =  1, 

Fy^  the  tangent^  and  P^N  the  normal  at  the  point 
Pi=(a^i,^i). 


162-163.] 


THE  HYPERBOLA 


269 


Then  the  equation  of  Pj^  is  ^  -  ^  =  1,  and  the  length  of 


^' 


the  intercept  OT  oi  the  tangent  is 


0T=-, 


Now,  in  the  triangle  F1P1F2, 


a' 


FiT=F,0  +  0T=  ae  +- 

Xi 


a 


and 


=  —(^exi  +  a), 

Xi 

TF,=  OF,-  0T  = 


a" 

ae 

a?! 


Xy 


(^exi  —  a); 


[Art.  161] 


but  FiPi  =  exi  4-  ^, 

and  PiF2  =  exi  —  a. 

Hence  F^T  :  TF^  =  F^P,  :  P1P2. 

and,  by  plane  geometry,  the  tangent  bisects  internally  the 
angle  between  the  focal  radii.  Then,  since  the  normal  is 
perpendicular  to  the  tangent,  the  normal  PiJV  bisects  the 
external  angle  F2P1W.     These  facts  suggest  a  method,  anal- 


270 


ANALYTIC  GEOMETRY 


[Ch.  XL 


ogous  to  that  of  Art.  148,  for  constructing  the  tangent  and 
normal  to  an  hyperbola  at  a  given  point. 

164.   Conjugate  hyperbolas.     A  curve  bearing  very  close 
relations  to  the  hyperbola 


=  1 


a" 


is  that  represented  by  the  equation 

"P  ~  ^  ""    ' 


i.e.,  by 


(1) 


(2) 


Fig. 116. 


in  which  a  and  h  have  the  same  values  as  in  equation  (1). 
This  curve  is  evidently  an  hyperbola,  and  has  for  its  trans- 
verse and  conjugate  axes,  respectively,  the  conjugate  and 
transverse  axes  of  the  original,  or  primary  hyperbola.  Two 
such  hyperbolas  are  called  conjugate  hyperbolas  ;  they  are 
sometimes  spoken  of  as  the  x-  and  y-hyperbolas,  respectively. 


163-164.]  THE  HYPERBOLA  271 

It  follows  at  once  that  the  hyperbola  (2),  conjugate  to 
the  hyperbola  (1),  has  for  its  eccentricity 


for  foci  the  points  (0,  ±be')^  and  for  directrices  the  lines 

b 

Two  conjugate  hyperbolas  have  a  common  center,  and 
their  foci  are  all  at  the  common  distance  Va^  -j.  J2  from  this 
center;  i.e.,  the  foci  all  lie  on  a  circle  about  the  center, 
having  for  radius  the  semi-diagonal  OS  of  the  rectangle 
upon  their  common  axes,  and  whose  sides  are  tangent  to  the 
curves  at  their  vertices.  Moreover,  when  the  curves  are 
constructed  it  will  be  found  that  they  do  not  intersect,  but 
are  separated  by  the  extended  diagonals  OS  and  OK  of  this 
circumscribed  rectangle,  which  they  approach  from  opposite 
sides.  These  diagonals  are  examples  of  a  class  of  lines  of 
great  interest  in  analytic  theory,  called  asymptotes  (cf .  Art. 
3T,  (0). 

EXERCISES 

1.  Construct  an  hyperbola,  given  the  distance  between  its  foci  as 
3  cm. 

2o  Construct  an  hyperbola,  given  the  distance  from  directrix  to  focus 
as  2  cm. 

3.  Write  the  equation  of  an  hyperbola  conjugate  to  the  hyperbola 
9x^  —  IQy^  =  144,  and  find  its  axes,  foci,  and  latus  rectum.  Sketch  the 
figure. 

4.  Write  the  equations  of  the  tangent  and  normal  to  the  hyperbola 
16a;2  _  9y2  _  j^i2  at  the  point  (4,  4),  and  find  the  subtangent  and  sub- 
normal. 

5.  Write  the  equations  of  the  polars  of  the  point  (3,  4)  with  respect 
to  the  hyperbola  Qx'^  —  IQy^  =  144  and  its  conjugate,  respectively. 


272  ANALYTIC  GEOMETRY  [Ch.  XI. 

6.  For  what  points  of  an  hj'perbola  are  the  subtangent  and  sub- 
normal equal  ? 

7.  Given  the  hyperbola  9  ?/2  —  4  x^  =  36,  find  the  focal  radii  of  the 
point  whose  ordinate  is  (  "1),  and  abscissa  positive. 

8.  A  tangent  which  is  parallel  to  the  line  5a:  —  4?/  +  7  =  0,  is  drawn 
to  the  hyperbola  x^  —  y'^=Q\  what  is  the  subnormal  for  the  point  of  con- 
tact? 

9.  What  tangent  to  the  hyperbola ^  =  1  has  its  ?/-intercept  2? 

10.  Find,  by  equation  [67],  the  two  tangents  to  the  hyperbola 
4:c2  —  2  2/2  zr  6  which  are  drawn  through  the  point  (3,  5). 

11.  Find  the  polars  of  the  vertices  of  an  hyperbola  with  respect  to  its 
conjugate  hyperbola. 

12.  Prove  that  if  the  crack  of  a  rifle  and  the  thud  of  the  ball  on  the 
target  are  heard  at  the  same  instant,  the  locus  of  the  hearer  is  an 
hj'perbola. 

13.  An  ellipse  and  hyperbola  have  the  same  axes.  Show  that  the 
polar  of  any  point  on  either  curve  is  a  tangent  to  the  other. 

14.  Find  the  equation  of  an  hyperbola  whose  vertex  bisects  the  dis- 
tance from  the  focus  to  the  center. 

15.  If  e  and  e'  are  the  eccentricities  of  an  hyperbola  and  its  conjugate, 
then 

e2  +  g'2  ^  g2g'2. 

16.  If  e  and  e'  are  the  eccentricities  of  two  conjugate  hyperbolas, 
then 

ae  =  he'. 

17.  Find  the  eccentricity  and  latus  rectum  of  the  hyperbola 

2/2  =  4(^2  +  a-). 

18.  Find  the  tangents  to  the  hyperbola  9  x^  —  16  ?/2  =  144,  which, 
with  the  tangent  at  the  vertex,  form  a  circumscribed  equilateral  triangle. 
Find  the  area  of  the  triangle. 

19.  Find  the  lengths  of  the  tangent,  normal,  subtangent,  and  sub- 
normal for  the  point  (3,  2)  of  the  hyperbola  x^  —  2y'^  =  1. 

165.  Asymptotes.  If  a  tangent  to  an  infinite  branch  of  a 
curve  approaches  more  and  more  closely  to  a  fixed  straight 
line  as  a  limiting  position,  when  the  point  of  contact  moves 
further  and  further  away  on  the  curve  and  becomes  infinitely 


164-165.] 


THE  HYPERBOLA 


273 


distant,  then  the  fixed  line  is  called  an  asymptote  of  the 
curve.*  More  briefly,  though  less  accurately,  this  defini- 
tion may  be  stated  as  follows  : 
an  asymptote  to  a  curve  is  a 
tangent  whose  point  of  contact 
is  at  infinity,  but  which  is  not 
itself  entirely  at  infinity.  It  is 
evident  that  to  have  an  asymp- 
tote a  curve  must  have  an  infi- 
nite branch ;  and  this  branch 
may  be  considered  as  having 
two  coincident,  and  infinitely 
distant,  points  of  intersection  with  its  asymptote.  This 
property  will  aid  in  obtaining  the  equation  of  the  asymptote. 


Y 

N.  ^V       N, 

^'z 

s/ 

\ 

■^   / 

F^ 

A'V 

"Va 

'  F. 

X 

/ 

/ 

B^ 

\ 

# 

Fl    ^ 

"% 

Fig.  117 


The  hyperbola 


X 


a' 


-2^=1 

79  -'-9 


(1) 

(2) 


is  cut  by  the  line  y  =  mx  +  <?, 

in  two  points  whose  abscissas  are  given  by  the  equation 

QaV-y'^x^ +  2a^cmx-^a:'h''-\-a'c''  =  0,   .     .     (3) 

If  line  (2)  is  an  asymptote,  the  two  roots  of  equation  (3) 
must  both  become  infinite  ;  therefore,  by  Art.  10, 


a?w? 


hence 


5^  =  0     and     2  a?cm  =  0, 

c  =  0     and     m=  ±-. 

a 


(4) 


Substituting  these  values  in  equation  (2),  gives 


y=-x. 


and    y  =  —-  x^ 


(5) 


*  This  definition  implies  that  the  distance  between  a  curve  and  its 
asymptote  becomes  infinitely  small.  McMahon  &  Snyder,  Differential  Cal- 
culus, Chap.  XIV. 

TAN.    AN.    GEOM.  —  18 


274  AI^ALYTIC  GEOMETRY  [Ch.  XI. 

and  these  equations  represent  the  asymptotes  of  the  hyper- 
bola ;  they  are  the  lines  OS  and  O^in  Fig.  117.  Therefore, 
the  hyperbola  has  two  asymptotes^  which  pass  through  its  center ; 
they  are  the  diagonals  of  the  rectangle  described  upon  its  axes. 

Since  the  equation  of  the  hyperbola  conjugate  to  (1)  is 

-o--o=-^^  •  •  •  (<5) 

a^      b^ 

and  thus  differs  from  equation  (1)  only  in  the  sign  of  the 
second  member,  which  affects  only  the  constant  term  in 
equation  (3),  therefore  the  equations  (4)  determine  the 
value  of  m  and  e  for  the  asymptotes  of  the  conjugate  hyper- 
bola also.  It  follows  that  coyijugate  hyperbolas  have  the  same 
asymptotes, 

A  second  derivation  of  the  equation  of  the  asymptotes  of  an  hyper- 
bola (1)  is  as  follows  : 

The  equation  of  the  tangent  to  (1)  at  the  point  (a;^,  y^  is 


which  may  be  written  in  the  form 


«^^f:+v   •   •  •   (8) 


Since  (x^,  y^)  is  on  the  curve  (1), 

therefore  -^-|^1,     i.e.,      |  =  aI5^-         •  ■  (9) 

Substituting  this  value  of  ^  in  equation  (8),  it  becomes 


b^x  =  ce-y\—^ -_  + ,         .  .         .         (10) 

a-^      x^^       x-^ 

which  is  only  another  form  of  the  equation  of  the  tangent  represented 
by  equations  (7)  or  (8).  If  now  the  point  of  contact  (a:^,  y-^  moves 
further  and  further  away,  so  that  a;^  =  co  ,  then  the  limiting  position  of 

the  line  (10)  is  represented  by  b'^x  =  a^y  (  -t  _  j  =  ±  aby. 

Hence  the  equations  of  the  asymptotes  are  :  y  =±  -x  (cf.  Art.  156). 


1G5-166.]  THE  HYPERBOLA  275 

The  equations  of  the  asymptotes  may  be  comhined,  by 
Art.  40,  into  the  one  equation  which  represents  both  lines, 


VIZ.: 


^-i^  =  0.        .         .         .         [68] 

166.  Relation  between  conjugate  hyperbolas  and  their 
asymptotes.  It  has  been  seen  that  the  standard  forms  for 
the  equations  of  the  primary  hyperbola,  its  asymptotes,  and 
its  conjugate  hyperbola  are,  respectively. 


a'      52 


x^      y^ 


a 


2      52 


0,  .         .         .         (2) 


/y>^  ft  lit 

^-|=-1.       ...         (3) 

It  will  be  noticed  at  once  that  these  three  equations  differ 
only  in  their  constant  terms ;  and  that  the  equation  of  the 
primary  hyperbola  (1)  differs  from  that  of  the  asymptotes 
(2)  by  the  negative  of  the  constant  by  which  the  equation 
of  the  conjugate  hyperbola  (3)  differs  from  equation  (2). 
Moreover,  this  relation  between  the  equations  of  the  three 
loci  must  hold  when  not  in  their  standard  forms,  z.6.,  Avhat- 
ever  the  coordinate  axes.  For,  any  transformation  of  coor- 
dinates will  affect  only  the  first  member  of  equations  (1), 
(2),  and  (3),  and  will  affect  these  in  precisely  the  same  way. 
After  the  transformation,  therefore,  the  equations  of  the  loci 
will  differ  only  by  a  constant  (not,  however,  usually  by  1); 
and  the  value  of  the  constant  in  the  equation  of  the 
asymptotes  will  be  midway  between  the  values  of  the  con- 
stants in  the  equations  of  the  two  hyperbolas. 


276  ANALYTIC  GEOMETRY  [Ch.  XI. 

Example  1.     An  hyperbola  having  the  lines 

(1)   2:  + 2^  + 3  =  0     and     (2)    3a; +  4^ +  5=0 
for  asymptotes,  will  have  an  equation  of  the  form 

(a;  +  2y  +  3)(3a;  +  4^  +  5)  +  A:  =  0,      .     .     (3) 
while  the  equation  of  its  conjugate  hyperbola  will  be 

(a;  +  2^  +  3)(32;  +  4y  +  5)-y^  =  0.      .     .      (4) 

If  a  second  condition  is  imposed  upon  the  hyperbola, 
e.g.^  that  it  shall  pass  through  the  point  (1,  "1),  then  the 
value  of  k  may  be  easily  found  thus :  since  tlie  curve  passes 
through  the  point  (1,  ~1),  therefore  by  equation  (3), 

(l_2  +  3)(3-4  +  5)  +  A;  =  0;    .-.  ^  =  -8, 
and  the  equation  of  the  hyperbola  is 

(x  +  2y  +  3)(3a;  +  4?/  +  5) -  8  =  0, 
that  is,         32;2  +  10:?;?/ +  8^2.^142;  + 22y  + 7  =  0;       .     (5) 
and  the  equation  of  the  conjugate  hyperbola  is 

2>x^  +  lOxy  +  8^2  ^  14a:  +  22^  +  23  =  0. 

Example   2.      The   equation  of   the  asymptotes  of  the 

hyperbola 

3a;2-14a;^-5i/2  +  7rr  +  13^-8  =  0   .    .    .    (1) 

differs  from  equation  (1)  by  a  constant  only,  hence  it  is  of 

the  form 

^x^-Uxy-by'^^-lx^-ny  +  k  =  0.    .     .     (2) 

Now  equation  (2)  represents  a  pair  of  straight  lines,  there- 
fore its  first  member  can  be  factored,  and,  by  Art.  67,  [17] 

-Wk-  ^^  _  5 0 i  +  24 5  _  49 ^  =  0  ; 
i.e.,  Q4:k  =  —  384,     whence     k  =  —  6, 

Therefore  the  equation  of  the  asymptotes  is 

Sx^-Uxy-5y^  +  lx  +  lSy-6  =  0, 
I.e.,  (3  a;  +  2/  -  2) (^  -  5  ?/  4-  3)  =  0  ; 


166-167.]  THE  HYPERBOLA  277 

and  the  equation  of  the  conjugate  hyperbola  is 

3  :r2  -  14a^^  -  5  ?/2  +  7:^;  -f  13  ^  -  4  =  0. 

167.   Equilateral  or  rectangular  hyperbola.     If  the  axes  of 
an  hyperbola  are  equal,  so  that  a  =  h,  its  equation  has  the 

foi*!^  x^-y'^  =  a\         .  .  .  (1) 

and  its  eccentricity  e  =  V2.  Its  conjugate  hyperbola  has 
the  equation  ^2_^2^_^2.      .         .         ,         (2) 

with  the  same  eccentricity  and  the  same  shape ;  while  its 
asymptotes  have  the  equations 

x=±y,  .         .         .         (3) 

and  are  therefore  the  bisectors  of  the  angles  formed  by  the 
axes  of  the  curves ;  hence  the  asymptotes  of  these  hyper- 
bolas are  perpendicular  to  each  other.  The  hyperbola  whose 
axes  are  equal  is  therefore  called  an  equilateral,  or  a  rec- 
tangular hyperbola,  according  as  it  is  thought  of  as  having 
equal  axes  or  asymptotes  at  right  angles. 

EXERCISE 

1.  Find  the  asymptotes  of  the  hyperbola  9  a:^  —  16  z/^  =  25,  and  the 
angle  between  them. 

2.  What  are  the  poles  of  the  asymptotes  of  the  hyperbola 

9a;2-  16?/2  =  25 
with  reference  to  the  curve  ? 

3.  If  the  vertex  lies  two  thirds  of  the  distance  from  the  center  to 
the  focus,  find  the  equations  of  the  hyperbola,  and  of  its  asymptotes. 

9  0 

4.  If  a  line   y  =  mx  +  c  meets   the    hyperbola  —  —  ^.^  =  1    in    one 
finite  and  one  infinitely  distant  point,  the  line  is  parallel  to  an  asymptote. 

5.  Show  that,  in  an  equilateral  hyperbola,  the  distance  of  a  point 
from  the  center  is  a  mean  proportional  between  its  focal  distances. 

6.  Find  the  equation  of  the  hyperbola  passing  through  the  point 
(0,  7),  and  having  for  asymptotes  the  lines 

2x  -y  =  7,  and  dx  +  3y  =  b  (cf.  Art.  166). 


278  ANALYTIC   GEOMETRY  [Ch.  XI. 

7.  Write  the  equation  of  the  hyperbola  conjugate  to  that  of  Ex.  6. 

8.  Find  the  equations  of  the  asymptotes  of  the  hyperbola 

2x^  —  xy  —  2x  =  y'^  -}-  y  -\-  Q'^ 
also  find  the  equation  of  the  conjugate  hyperbola. 

9.  Find  the  equation  of  the  asymptotes  of  the  hyperbola 

dx^  +  34:xy  +  Uy^  -  X  +  21y  =  0. 

10.  Find  the  equation  of  the  hyperbola  conjugate  to 

9^2  _  16^2  _^  36^^,  ^  iQo^  ^  508. 

11.  Prove  that  a  perpendicular  from  the  focus  to  an  asymptote  of  an 
hyperbola  is  equal  to  the  semi-conjugate  axis. 

12.  The  asymptotes  meet  the  directrices  of  the  x-hyperbola  on  the 
a:-auxiliary  circle,  and  of  the  conjugate  hyperbola  on  the  ^/-auxiliary  circle. 

13.  The  circle  described  about  a  focus,  with  a  radius  equal  to  half  the 
conjugate  axis,  will  pass  through  the  intersections  of  the  asymptotes 
and  a  directrix. 

14.  The  line  from  the  center  C  to  the  focus  F  of  an  hyperbola  is  the 
diameter  of  a  circle  that  cuts  an  asymptote  at  P;  show  that  the  chords 
CP  and  FP  are  equal,  respectively,  to  the  semi-transvei"se  and  semi- 
conjugate  axes. 

168.  The  hyperbola  referred  to   its   asymptotes.      If  the 

asymptotes  of  an  hyperbola  are  chosen  as  the  coordinate 
axes,  their  equations  will  be  2:  =  0  and  «/  =  0,  respectively  ; 
or,  combined  in  one  equation, 

x^  =  0,  .  .  .  (1) 

By  the  reasoning  of  Art.  166,  it  follows  that  the  equation 
of  the.  hyperbola,  —  which  differs  from  that  of  its  asymptotes 
by  a  constant,  —  is 

x^  =  Jc,  .  .         •  .  (2) 

wherein  the  value  of  the  constant  7c  is  to  be  determined  by 
an  additional  assigned  condition  concerning  the  curve  j  e.g., 
that  it  shall  pass  through  a  given  point. 


167-168.] 


THE  HYPERBOLA 


279 


The  value  of  this  constant,  in  terms  of  a  and  6,  can  in 
general  be  found  most  easily  by  making  the  proper  trans- 
formation of  coordinates  upon  the  equation  of  the  hyperbola 


(3) 


Fig. 118. 


The  new  :r-axis  makes  the  angle  6,  the   new   ?/-axis   the 
angle  6\  with  the  old  a;-axis,  such  that 


tan  6  = •»    tan  0'  =-- 

a  a 


Hence 
and 


sm 


e  =  -^ine'  = 


-h 


■y/a?  +  52 


cos  6  =  +  cos  6'  = 


a 


Va2  +  52 

therefore  the  formulas  [24]  for  transformation, 

x  =  x'  cos  ^  +  ?/'  COS  ^^     ^  =  x'  sin  0  +  y'  sin  6\ 
become  in  this  case 


x  = 


Va2  +  62 


(x'  +  y ),     y  = 


Va2  +  62 


(:r'-y), 


(4) 


280  ANALYTIC  GEOMETRY  [Ch.  XI. 

Appljdng  this  transformation,  equation  (3)  becomes 

x'^  +  2  x'y'  +  y'^     x'^  -  2  x'y'  +  y'^  ^  ^  , 
a^  -\-  b"^  G^  -\-W' 

that  is,  dropping  the  accents, 

^y^<l^^  .  .  .  [69] 

which  is  the  desired  equation  of  the  hyperbola  when  referred 
to  its  asymptotes  as  coordinate  axes. 

The  equation  of  the  conjugate  hyperbola  is  then 

-^  +  ^^      ...         (5) 


xy  = 


4 


Remembering  the  relation  IP"  =  c^{(^  —  X)^  it  will  be  seen 

that  the  value  of  the  constant  term  in  equation  (2)  may  be 

written 

7  _  a^  +  5^  _  G^^  _   o 

*      ^-^    '^' 

so  that  G  is  half  the  distance  of  the  focus  from  the  center  of 
the  curve.  Again,  the  coordinates  of  the  foci,  a;  =  ±  ae,  ^  =  0, 
become  after  the  transformation  (4), 

x  =  y=±—- ;        ...        (6) 

z  a 

and  the  equations  of  the  directrices,  x  =  ±-,  become 

e 

X  +  y  =  ±a.         .         .         .         (7) 

169.   The  tangent  to  the  hyperbola  ocy  =  c^.     The  equation 
of  the  tangent  to  the  hyperbola 

xy  =  c\  .  .  .  (1) 

at  any  given  point  (a^i,  y^),  may  be  easily  derived  by  the 
secant  method  (cf.  Arts.  84,  122).  Let  Pi=(xi,  y{)  and 
^2=  (^2'  ^2)  ^®  ^^^  points  on  the  curve  ;  then 

^ij/i  =  ^1       •       •       (2)    and    x^^  =  <?'.      .       .       (3) 


168-170.]  THE  HYPERBOLA  281 

The  equation  of  the  line  through  P^  and  P^  is 

wherein  ^^  ~  ^^  must  have  a  value  determined  by  equations 
(2)  and  (3),  hence 

ley  JU-i  V  Uy-t  ^2 


m  = 


Xsy    iC-l  X-iXn       Xsy  X-l  X-lXty^ 

The  equation  of  the  secant  line  P^Pg  i*^  therefore 

y  -y    =  -  J—-(x-x^^.       ...       (4) 

If  now  the  point  P^  becomes  coincident  with  P^,  equation 
(4)  becomes 

^-^1=  -—2(^-^1)' 
which  may  be  reduced  by  equation  (2)  to 

-  +^  =  2,         .         .         .         [70] 

^1    yi 

or  to  y-^x  +  2;^^  =  2  c^, 

which  is  the  required  equation  of  the  tangent  at  the  point 
P^=(x^,  y^  of  the  curve. 

170.  Geometric  properties  of  the  hyperbola.  Equation  [69] 
states  the  following  intrinsic  property  for  the  hyperbola, 
P^=(xj,  y-^  being  any  point  on  the  curve  (Fig.  119). 

4  MP^  •  XPi  =  OF  ; 

that  is,  for  every  point  of  the  hyperbola^  four  times  the  product 
of  its  distances  from  the  asymptotes^  measured  parallel  to  the 
asymptotes  respectively^  is  equal  to  the  square  of  the  distance 
from  the  center  to  the  focus ;  and  is  therefore  constant. 


ANALYTIC  GEOMETRY 


[Ch.  XI. 


Again,  26  being  the  angle  between  the  asymptotes,  equa- 
tion [69]  may  be  written 


xy  sm  2 ^= ^ ^ —  sm  2  6. 

^  2  2 


•  (I) 


Fig. 119. 


Now   xy  sin  2  ^   is   the  area  of  the  parallelogram   OMP^L, 

constructed   upon  the  coordinates  of  the   point  P^   of   the 

hyperbola ;    and  since  the  coordinates  of  the  vertex  A  are 

Va^  4-  ^2 
X  —  y  = — — ,  the  second  member  of  equation  (1)  is  the 

area  of  the  rhombus  OR  AS,  constructed  upon  the  coordinates 
of  the  vertex.  Therefore,  the  area  of  the  parallelogram 
formed  by  the  asymptotes  and  lines  parallel  to  them  drawn 
from  any  point  of  an  hypei^hola,  is  constant;  it  is  equal  to 
the  rhombus  similarly  drawn  from  the  vertex  of  the  curve. 
The  equation  of  the  tangent  to  the  hyperbola 

xy  =  c^.,  .  .  .  (2) 

at  the  point  P,  is  ^  + 1^  =  2.        .  .      ^    .  (3) 

The  a^-intercept  of  this  tangent  is  OT  —  'lx^\  hence  if  OT 
be  the  y-intercept,  and  M  the  foot  of  the  ordinate  of  Pj, 
then  from  the  similar  triangles  MTP^  and  OTT\ 


TP^  :  TT'  =  MT :  OT  = 


*//-i 


2x^  =  1:2. 


170.]  THE  HYPEBBOLA  283 

Hence,  the  segment  of  any  tangent  to  an  hyperhola  between 
the  asymptotes  is  bisected  by  the  point  of  contact. 

The  tangent  (3)  has  the  intercepts  on  the  a;-axis  and  ^-axis, 
respectively, 

Then  OT'  OT'  =  4.x^y^,      ...        (4) 

But  since  (rr^,  y^  is  a  point  of  the  hyperbola 

4  x-^y-^  =  a^  -\-  h\ 

hence  OT-  OT  =a^  -^h\      ,         .        .        (5) 

i.e.,  the  rectangle  formed  hy  the  intercepts  ichich  any  tangent 
to  the  hyperbola  makes  upon  the  asymptotes  is  constant;  it  is 
equal  to  the  sum  of  the  squares  upon  the  semi-axes. 

Moreover,  equation  (5)  may  be  written 


but      sin  2^  =  2  sin  Q  cos  Q 


...     (6) 

b  a  2  ab 


V«2  +  z^:^  Va'^4-62      o^  +  b"^ 

OT '  OT' 

hence  (6)  bcomes sin  2  ^  =  ab-^       ....     (4). 

that  is,  the  triangle  formed  by  any  tangent  to  an  hyperbola 
and  its  asymptotes  is  constant;  it  is  equal  to  the  rectangle 
upon  the  semi-axes. 

EXERCISES 

1.  Find  the  equation  of  the  hyperbola  9x^  —  IQy^  =  25  when  referred 
to  its  asymptotes  as  axes. 

2.  Find  the  semi-axes,  eccentricity,  and  the  vertices,  of  the  hyperbola 
xy  =  4,  the  angle  between  the  axes  (asymptotes)  being  90°. 

3.  Find  the  semi-axes,  eccentricity,  vertices,  and  the  foci,  of  the  hyper- 
bola xy  =  —12,  the  angle  between  the  axes  being  G0°. 

4.  Prove  that  the  segments  of  any  line  which  are  intercepted  between 
an  hyperbola  and  its  asymptotes  are  equal. 


284  ANALYTIC   GEOMETRY  [Ch.  XL 

5.  Express  the  angle  between  the  asymptotes  of  an  hyperbola  in  terms 
of  e ;  i.e.,  in  terms  of  the  eccentricity  of  the  hyperbola. 

6.  The  segment  of  a  tangent  to  an  hyperbola  intercepted  by  the 
conjugate  hyperbola  is  bisected  at  the  point  of  contact. 

7.  Show  that  the  pole  of  any  tangent  to  the  rectangular  hyperbola 
xy  =  c^,  with  respect  to  the  circle  x^  ■\-  ip-  —  a^,  lies  on  a  concentric  and 
similarly  placed  rectangular  hyperbola. 

8.  Prove  that  the  asymptotes  of  the  hyperbola  xy  =  hx  -\-  ky  are 
X  =  k,  and  y  —  li. 

9.  Derive  the  equation  of  the  tangent  to  the  curve  xy  =  hx  -{■  ky  at 
the  point  P  =  (a:^,  y^)  on  the  curve. 

171.  Diameters.  A  diameter  has  already  been  defined 
(Art.  129)  as  the  locus  of  the  middle  points  of  a  system  of 
parallel  chords,  and  in  Art.  152  the  equation  was  derived 
for  a  diameter  of  an  ellipse.  By  the  same  method,  if  a  sys- 
tem of  parallel  chords  of  the  hyperbola 

have  the  common  slope  m,  the  equation  of  the  corresponding 
diameter  will  be  found  to  be 

y  =  -^x,         .  .         .  [71] 

This  equation  shows  that  every  diameter  of  the  hyperbola 
passes  through  the  center. 

Conversely,  it  is  true,  as  in  the  case  of  the  ellipse,  that 
every  chord  of  the  hyperbola  through  the  center  is  a  diame- 
ter. That  chord  of  the  original  set  which  passes  through 
the  center  is  the  diameter  conjugate  to  [71] ;  and  its  equa- 
tion is 

y  =  mx.        .  .  .         [72] 


171-172.]  THE  HYPERBOLA  285 

Letting  m'  be  the  slope  of  a  diameter,  and  m  that  of  its 
conjugate,  the  essential  condition  that  two  diameters  should 
be  conjugate  to  each  other  is  that  (cf.  Art.  153) 

mm'  =  — .        .  .  .  [73] 

(Ju 

172.   Properties  of   conjugate   diameters  of  the  hyperbola. 

(a)  It  is  clear  that  the  condition 


mm'  =  —         .         ,         ,  [731 


holds  also  for  the  hyperbola 

which  is  conjugate  to  the  given  hyperbola ;  for,  replacing  a^ 
by  —  a^  and  —  P  by  b^  leaves  equation  [73]  unchanged. 
Hence,  diameters  which  are  conjugate  to  each  other  for  a  given 
hyperbola  are  conjugates  also  for  the  conjugate  of  that  hyper- 
bola. 

(/3)  The  axes  of  the  hyperbola  are  clearly  diameters  of 
the  curve.  They  are  perpendicular  to  each  other,  and 
therefore  satisfy  the  relation 

mm'  =  —  1. 

Comparing  this  condition  with  that  of  equation  [73],  it 
follows  that  the  transverse  and  conjugate  axes  of  the  hyper- 
bola are  the  only  pair  of  perpendicular  conjugate  diameters 
(cf.  C/3)  p.  255). 

li  a  =  b^  the  condition  [73]  reduces  to 

m7n'  =  1  ; 

therefore  (Art.  16),  in  the  rectangular  hyperbola  the  sum 
of  the  angles  which  a  pair  of  conjugate  diameters  make 
with  the  transverse  axis  is  90°  (cf.  Art.  156). 


286  ANALYTIC  GEOMETRY  [Cii.  XL 

(7)  Since  in  equation  [73]  the  product  mm'  is  positive, 
it  follows  that  the  angles  which  conjugate  diameters  make 
with  the  transverse  axis  are  both  acute,  or  both  obtuse. 
Moreover, 

if  m  <  ±  -,  then  711'  >  ±  -  ; 
a  a 

and  the  diameters  lie  on  opposite  sides  of  an  asymptote. 
Two  conjugate  diameters  lie  in  the  same  quadrant  formed  hy 
the  axes  of  the  hyperbola^  on  opposite  sides  of  the  asymptote 
(cf.  Art.  155  (a)). 

(S)  An  asymptote  passes  through  the  center  of  an  hyper- 
bola, hence  may  be  regarded  as  a  diameter.     Its  slope  is 

m=  ±  -,         .*.  m  =  ±-\ 
a  a 

hence,  an  asymptote  regarded  as  a  diameter  is  its  oivn  conju- 
gate ;  it  may  be  called  a  self-conjugate  diameter. 

This  is  a  limiting  case  of  (7)  above. 

(e)  It  follows  from  this  last  fact  that  if  a  diameter  inter- 
sects a  given  hyperbola,  then  the  conjugate  diameter  does 
not  intersect  it,  but  cuts  the  conjugate  hyperbola.  It  is 
customary  and  useful  to  define  as  the  extremities  of  the 
conjugate  diameter  its  points  of  intersection  with  the  conju- 
gate hyperbola.  With  this  limitation,  it  follows  from  (a) 
of  this  article,  that,  as  in  the  ellipse,  each  of  two  conjugate 
diameters  bisects  the  chords  parallel  to  the  other. 

(^)  As  a  limiting  case  of  this  last  proposition,  also,  it  is 
evident  that  the  tangent  at  the  end  of  a  diameter  is  parallel 
to  the  conjugate  diameter. 

By  reasoning  entirely  analogous  to  tliat  given  in  Art.  155, 
for  the  ellipse,  properties  similar  to  those  there  given  may 
be  derived  for  the  hyperbola.  They  are  included  in  the 
following  exercises,'  to  be  worked  out  by  the  student. 


172-173.]  THE  UYPEEBOLA  287 

EXERCISES 

1.  Find  the  equation  of  the  diameter  of  the  hyperbola 

9  x2  -  16  ?/2  =  25 
which  bisects  the  chords        y  =  ?>  x  -\-  h. 
Find  also  tlie  conjugate  diameter. 

2.  Find,  for  the  liyperbola  of  Ex,  1,  a  diameter  through  the  point 
(1,  1),  and  its  conjugate. 

X         ifi 

3.  Find  the  diameter  of  the  hyperbola  ^^  ~  nH  =  1  which  is  con- 
jugate to  the  diameter  x  —  oy  =  0. 

4.  Find  the  equation  of  a  chord  of  the  hyperbola  12  x^  —  9  ?/^  =  108, 
which  is  bisected  at  the  point  (4,  2). 

5.  Lines  from  any  point  of  an  equilateral  hyperbola  to  the  extremi- 
ties of  a  diameter  make  equal  angles  with  the  asymptotes. 

6.  Show  that,  in  an  equilateral  hyperbola,  conjugate  diameters  make 
equal  angles  with  the  asymptotes. 

7.  The  difference  of  the  squares  of  two  conjugate  semi-diameters  is 
constant ;  it  is  equal  to  the  difference  of  the  squares  of  the  semi-axes. 

8.  The  angle  between  two  conjugate  diameters  is  sin-^-^. 

a'h' 

9.  The  polar  of  one  end  of  a  diameter  of  an  hyperbola,  with  reference 
to  the  conjugate  hyperbola,  is  the  tangent  at  the  other  end  of  the 
given  diameter. 

10.  Tangents  at  the  ends  of  a  pair  of  conjugate  diameters  intersect 
on  an  asymptote. 

173.  Supplemental  chords.  As  previously  defined,  chords  of  a  curve 
are  supplemental  when  drawn  from  any  point  of  the  curve  to  the  ex- 
tremities of  a  diameter.  If,  in  the  analytic  work  of  Art.  157,  V^  is 
replaced  by  —y^,  then,  if  m  and  m'  are  the  slopes  of  two  supplemental 
chords  of  the  hyperbola,  they  must  satisfy  the  relation 

mm  =  — .  .  .  .  (1) 

But  this  is  (see  Eq.  [73])  the  condition  that  exists  between  the 
slopes  of  two  conjugate  diameters.  Therefore,  supplemental  chords  are 
parallel  to  a  pair  of  conjugate  diameters. 

For  the  equilateral  hyperbola,  i.e.,  when  a  =  b,  this  relation  has  the 
special  value  mm' =  1,  ...  (2) 


288 


A^^ALYrIC  GEOMETRY 


[Ch.  XL 


and,  therefore,  the  sum  of  the  acute  angles  which  a  pair  of  supplementary 
chords  of  the  equilateral  hyperbola  make  with  its  transverse  axis  is  90° 
(cf.  Art.  172  (/?)). 

174.  Equations  representing  an  hyperbola,  but  involving  only  one 
variable. 

(a)  Eccentric  angle.  In  the  theory  of  the  hyperbola,  the  auxiliary 
circles  described  upon  the  axes  of  the  curve  as  diameters  are  not  as 
useful  as  the  corresponding  circles  for  the  ellipse,  since  the  ordinate  for 
a  point  on  the  hyperbola  does  not  cut  the  a:-auxiliary  circle,  and,  there- 
fore, there  is  no  simple  construction  for  the  eccentric  angle.  It  is,  how- 
ever, sometimes  desirable  to  express  by  means  of  a  single  variable  the 
condition  that  a  point  shall  be  on  an  hyperbola;  and  for  this  purpose 
the  equations 

X  =  a  sec  (f),  y  =  b  tan  <f),         .         .        ,         [74] 

similar  to  equations  [60],  may  be  used;  for  these  evidently  satisfy  the 
equation  of  the  hyperbola 

a?-      b^        ' 


smce 


sec2<j!)  —  tan^^  =  1. 


The  angle  <^  may  be  defined  as  the  eccentric  angle  for  the  hyperbola, 
and  the  corresponding  point  of  the  curve  may  be  constructed  as  follows : 


Fig.  120. 


Draw  the  auxiliary  circles,  and  any  ZAOQ  —  <^.  At  the  points  R  and  Q, 
where  the  terminal  side  of  <^  cats  the  circles,  draw  tangents  cutting  the 
transverse  axis  in  the  points  M'  and  M,  respectively.     Erect  at  M  an 


173-174.]  THE  HYPEEBOLA  289 

ordinate  MP  equal  to  RM' ;  its  extremity  P  is  a  point  of  the  hyperbola. 
For,  in  the  right  triangle  OMQ,  ^ 

OM  cos  <}>  =0Q,      i.e.,  OM  =  a  sec  <^ ; 
and,  in  the  right  triangle  OM'R, 

M'R  =  OR  tan  </>,       i.e.,  M'R  =  b  tan  (jy. 

But  for  the  point  P, 

x=OM,  y  =  MP  =  M'R; 

hence  x  =  a  sec  <^,.y  =  h  tan  <^, 

and  P  is  a  point  on  the  hyperbola.* 

The  eccentric  angle  for  any  given  point,  P,  of  an  hyperbola  is  easily 
obtained.  Draw  the  ordinate  MP,  and  from  its  foot,  M,  draw  a  tangent 
MQ  to  the  a:-auxiliary  circle ;  then  the  angle  MOQ  is  the  eccentric  angle 
corresponding  to  P. 

(/?)   The  equation  of  the  hyperbola  referred  to  its  asymptotes,  viz. 

xy  =  c^,  is  satisfied  by  the  coordinates  x  =  ct,  y  =  -,  whatever  the  values 

of  t.    The  use  of  this  single  independent  variable  t  is  sometimes  convenient 
in  dealing  with  points  on  the  hyperbola.* 

EXAMPLES    ON    CHAPTER    XI 

1.  Write  the  equation  of  an  hyperbola  whose  transverse  axis  is  8, 
and  the  conjugate  axis  one  half  the  distance  between  the  foci. 

2.  Find  tBe  equation  of  that  diameter  of  the  hyperbola  lQx^  —  9y^  =  14:4: 
which  passes  through  the  point  (5,  J^) ;  also  find  the  coordinates  of  the 
extremities  of  the  conjugate  diameter. 

3.  Assume  the  equation  of  the  hyperbola,  and  show  that  the  difference 
of  the  focal  distances  is  constant. 

4.  Find  the  locus  of  the  vertex  of  a  triangle  of  given  base  2  c,  if  the 
difference  of  the  two  other  sides  is  a  constant,  and  equal  to  2  a. 

5.  Find  the*  locus  of  the  vertex  of  a  triangle  of  given  base,  if  the 
difference  of  the  tangents  of  the  base  angles  is  constant. 

6.  Find  an  expression  for  the  angle  between  any  pair  of  conjugate 
diameters  of  an  hyperbola. 

7.  Show  that  two  concentric  rectangular  hyperbolas,  whose  axes 
meet  at  an  angle  of  45°,  cnt  each  other  orthogonally. 

*  The  forms  of  this  article  are  useful  in  the  differential  calculus. 

TAN.    AN.    GEOM. 19 


290  ANALYTIC  GEOMETRY  [Ch.  XL 

8.  The  portions  of  any  chord  of  an  hyperbola  intercepted  between 
the  curve  and  its  conjugate  are  equal. 

Suggestion.     Draw  a  tangent  parallel  to  the  line  in  question. 

9.  The  coordinates  of  a  point  are  a  tan  (0  +  a)  and  b  tan  (0  +  (3) ', 
prove  that  the  locus  of  the  point,  as  6  varies,  is  an  hyperbola. 

10.  Prove  that  the  asymptotes  of  the  hyperbola  a:^  =  3  a:  +  5j/  are 
X  =  5  and  ^  =  3. 

11.  If  the  coordinate  axes  are  inclined  at  an  angle  w,  find  the  equa- 
tion of  an  hyperbola  whose  asymptotes  are  the  lines  x  =  2  and  ^  =  —  3, 
respectively,  and  which  passes  through  the  point  (2,  1). 

12.  Find  the  coordinates  of  the  points  of  contact  of  the  common 
tangents  to  the  hyperbolas, 

a;2  —  y2  _  3  ^2^   ^^^^  ^y  —  2a\ 

13.  If  a  right-angled  triangle  be  inscribed  in  a  rectangular  hyperbola, 
prove  that  the  tangent  at  the  right  angle  is  perpendicular  to  the 
hypothenuse. 


14.  Show  that  the  line  y  =  mx  -{•  2  kV -  jn  always  touches  the  hyper- 
bola xy  =  k^',  and  that  its  point  of  contact  is  (  ,  cV-  m) . 

15.  Find  the  point  of  the  hyperbola  xy  =  12  for  which  the  subtangent 
is  4.     Find  the  subnormal  for  the  same  point. 

16.  Find  the  polar  of  the  point  (5,  3)  on  the  hyperbola  x^  —  2y'^  =  7^ 
with  respect  to  the  conjugate  hyperbola.  Show  that  this  line  is  tangent 
to  the  given  hyperbola,  at  the  other  end  of  the  diameter  from  (5,  3). 

17.  If  an  ellipse  and  hyperbola  have  the  same  foci,  they  intersect  at 
right  angles. 

18.  Find  tangents  to  the  hyperbola  2y^  —  IQx^  =  1  which  are  perpen- 
dicular to  its  asymptotes.  • 

19.  Find  normals  to  the  hyperbola  ^^  ~  ^ v/  ~  -^)    =  1  which  are 

-^^  16  9 

parallel  to  its  asymptotes.     Find  the  polar  of  their  point  of  intersection. 

20.  Show  that,  in  an  equilateral  hyperbola,  conjugate  diameters  are 
equally  inclined  to  the  asymptotes. 

21.  Show  that  two  conjugate  diameters  of  a  rectangular  hyperbola 
are  equal. 


174.]  THE  HYPERBOLA  291 

22.  Show  that,  in  an  equilateral  hyperbola,  two  diameters  at  right 
angles  to  each  other  are  equal.     Show  also  that  this  follows  from  Ex.  21. 

23.  Find  the  sum  of  two  focal  chords  which  are,  respectively,  parallel 
to  two  conjugate  diameters. 

24.  Find  the  common  tangents  to  the  hyperbola .  =  1   and  its 

mid-circle  x^  -\-  y^  =  ab.  ^ 

25.  In  the  hyperbola  25  x^  —  16  y^  =  400,  find  the  conjugate  diameters 
that  cut  each  other  at  an  angle  of  45°. 

26.  The  latus  rectum  of  an  hyperbola  is  a  third  proportional  to  the 
two  axes. 

27.  The  polars  of  any  point  (h,  k)  with  respect  to  conjugate  hyperbolas 
are  parallel. 

28.  The  sum  of  the  eccentric  angles  of  the  extremities  of  two  conju- 
gate diameters  of  an  hyperbola  is  equal  to  90°;  i.e.,  cfi  -\-  cf)'  =  90°. 

29.  Find  the  equation  of  a  line  through  the  focus  of  an  hyperbola 
and  the  focus  of  its  conjugate,  and  find  the  pole  of  that  line. 

30.  Find  the  asymptotes  of  the  hyperbola  xy  —  3x  —  27/  =  0.  What 
is  the  equation  of  the  conjugate  hyperbola? 

31.  Show  that  the  ?/-axis  is  an  asymptote  of  the  hyperbola 

2xy  +  3x^-\-4tx  =  9. 

What  is  the  equation  of  the  other  asymptote?  Of  the  conjugate 
hyperbola  ? 

32.  If  two  tangents  are  drawn  from  an  external  point  to  an  hyperbola, 
they  will  touch  the  same  or  opposite  branches  of  the  curve  according  as 
the  given  point  lies  between  or  outside  of  the  asymptotes. 


CHAPTER   XII 

GENERAL  EQUATION  OF  THE   SECOND  DEGREE 

A3c'^  +  2  Hxij  +  By-  +  2  Gac  +  2  Fy  -{-  C  =  0 

175.  General  equation  of  the  second  degree  in  two  variables. 
Thus  far  only  special  equations  of  the  second  degree  have 
been  studied  ;  they  have  all  been  of  the  form 

Ax'^  +  Bi^^-\-2Gx  +  2Fi/+C=0,    .     .     .     (1) 

^.e.,  they  have  been  free  from  the  term  containing  the 
product  of  the  variables.  In  Arts.  107,  113,  and  119  it  is 
shown  that  equation  (1)  represents  a  conic  section  having 
its  axes  parallel  to  the  coordinate  axes.  It  still  remains  to 
be  shown,  however,  that  the  most  general  equation  of  the 
second  degree,  viz. 

Ax^-\-2ffx7/-{-Bf  +  2ax-{-2Fi/-hO=0,   ...     (2) 

also  represents  a  conic  section.  To  prove  this  it  is  only 
necessary  to  show  that,  by  a  suitable  change  of  the  coordi- 
nate axes,  equation  (2)  may  be  reduced  to  the  form  of 
equation  (1). 

If  equation  (2)  be  referred  to  new  axes,  OX'  and  OY'^ 
say,  making  an  angle  0  with  the  corresponding  given  axes; 
and  if  the  new  coordinates  of  any  point  on  the  curve  be  x' 
and  3/',  the  old  coordinates  of  the  same  point  being  x  and  ^ ; 
then  (Art.  T2) 

x  =  x'  cos  6  —  y'  sin  ^,  and  y  =  x'  sin  0  -\-  y'  cos  6.  .     .     (3) 

292 


Ch.  XII.  175.]      EQUATION  OF  SECOND  DEGREE  '       293 

Substituting  these  values  (3)  in  equation  (2),  it  becomes 

A(x^  cos  0  —  2/'  sin  Oy  +  2  R(x'  cos  O  —  y'  sin  ^) (x'  sin  6'  +  ?/'  cos  ^) 
+  ^(a;'  sin  6  +  y'  cos  (9)2  +  2  (r(a;'  cos  6'  -  ?/'  sin  0) 
-\-2F(x'  sin  (9  4-  ?/'  cos  6')  +  C  =  0,    .         .         .         (4) 

which,  being  expanded  and  re-arranged,  becomes  : 

^'2(^  cos2  ^  +  2  ^sin  ^  cos  ^  +  5  sin2  ^) 

+  ^'y'(-2^sin6>cos6'-2^sin2^  +  2^cos2  6'4-25sin^cos6>) 

+  y'XA  sin2  0_2  ^sin  Oco^e  +  B  cos2  ^) 

-l-a;'(2(7cos6'  +  2^sin(9) 

+  y(-2(7sin(9  +  2^cos6')+ (7=0.       ...         (5) 

This  transformed  equation  (5)  will  be  free  from  the  term 
containing  the  product  x'y'  if  6  be  so  chosen  that 

-2  A  sin  ^  cos  ^  -  2  irsin2  6*  +  2  iT'cos^  ^  -f  2  5sin  ^  cos6>  =  0, 

i.e.,  if         2  ^(cos2  0  -  sin2  (9)  =  (^  -  J5)2  sin  6  cos  ^, 

z.e.,  if  2  ^.  cos  2  (9  =  (^-^)  sin  2  6*, 

9  TT 

or  finally,  if  tan  26  =  — ^ — -  •       .        .        .        (6) 

Moreover,  it  is  always  possible  to  choose  a  positive  acute 
angle  6  so  as  to  satisfy  this  last  equation  whatever  may  be 
the  numbers  represented  by  JL,  B,  and  IT. 

Having  chosen  6  so  as  to  satisfy  equation  (6),  and  having 
substituted  the  values  of  sin  6  and  cos  0  in  equation  (5), 
that  equation  reduces  to 

A'x'^  +  B'l/^  +  2  a'x'  4-  2  ^',y '  +(7=0,.     .     (7) 

(wherein  A\  B\  •••  represent  the  new  coefficients) 

and  therefore  represents  a  conic  section  with  its  axes  parallel 
to  the  new  coordinate  axes.     But  equation  (7)  represents 


294  ANALYTIC  GEOMETRY  [Ch.  XII. 

the  same  locus  as  equation  (2);  hence  it  is  proved  that,  in 
rectangular  coordinates,  every  equation  of  the  form 

represents  a  conic  section  whose  axes  are  inclined  at  an  angle  6 

to    the  give7i   coordinate  axes,  zvhere  6  is  determined   hy  the 

equation  ^         9  tt 

^  taii20=    "^ 


A-B 

It  is  to  be  noted  that  the  constant  term  C  has  remained 
unchanged  by  the  transformation  given  above. 

The  next  article  will  illustrate  the  application  of  this 
method  to  numerical  equations.  It  is  to  be  observed  that 
this  method  is  entirely  general,  and  enables  one  to  fully 
determine  the  conic  represented  by  any  given  numerical 
equation  of  the  second  degree. 

Note.  In  the  proof  just  given  that  every  equation  of  the  second 
degree  represents  a  conic  section,  it  is  assumed  that  the  given  axes  are  at 
right  angles.  This  restriction  may,  however,  be  removed ;  for  if  they  are 
not  at  right  angles,  a  transformation  jnay  be  made  to  rectangular  axes 
having  the  same  origin  (cf.  Arts.  T-l,  75),  and  the  equation  will  have  its 
form  and  degree  left  unchanged;  after  which  the  proof  already  given 
applies. 

176.    Illustrative  examples.     Example  1.     Given  the  equation 

-a:2  +  4:r?/-2/2-4:  V2x  +  2  V2?/- 11  =0,     .     .     .     (1) 

to  determine  the  nature  and  position  of  its  locus. 

Turn  the  axes  through  an  angle  6,  i.e.,  substitute  for  x  and  y,  respec- 
tively, x'  cos  9  —  y'  sin  6  and  x'  sin  0  -\-  y'  cos  6]  equation  (1)  then  becomes 

x'2(  -cos2  ^  +  4  sin  ^  cos  0  -  sin2  6) 

+  x'y'{  +  2  sin  ^  cos  ^  +  4  cos^  ^  -  4  sin2  ^  -  2  sin  ^  cos  0) 

-  2/'2  (sin2  ^  +  4  sin  ^  cos  ^  +  cos2  0) 

-  a;'  (4  \/2  cos  0  -2\/2  sin  0)  _ 

+  y(+4\/2sin^  +  2  V2cos(9)-  11  =0.        ...       (2) 


175-176.] 


EQUATION    OF  SECOND  DEGREE 


295 


The  coefficient  of  x'y'  in  equation  (2)  reduces  to  4  (sin^  0  —  cos^  6)  ;  it 
will  therefore  be  zero  if  sin  6  =  cos  6,  i.e.,  \i  0  =  45°.* 

K  ^  =  45°,  then  sin  0  =  cos  6  =  - — -,  and  this  value  of  sin  0  and  cos  0 

V2 
being  substituted  in  equation  (2),  it  becomes 


3ij'2-2x'  +  6y  -11  =  0, 


(3) 


■which  represents  the  same  locus  as  is  represented  by  equation  (1)  ;  the 
difference  in  the  form  of  the  two  equations  bemg  due  to  the  fact  that  the 
axes  to  which  equation  (3)  is  referred  make  an  angle  of  45°  with  the  axes 
to  which  equation  (1)  is  referred. 

Equation  (3)  may  be  written  in  the  form 


(x'-iy-3(y'-iy  =  9, 


I.e., 


(x'-iy    (y'-iy 

(V3)^ 


32 


=  1, 


(4) 


which  represents  an  hyperbola  (cf.  Art.  118).  Its  center  is  at  the  point 
(1,  1) ;  the  transverse  axis  is  parallel  to  the  x'-axis;  the  semi-axes  are  of 
length  3  and  V3,  respec- 
tively; the  eccentricity  is  ^ 
e  =  |V3;  the  foci  are  at 
the  points  F=(l  -t-2  V3, 1) 
andF'  =  (l-2V3,  1),  re- 
spectively ;  the  directrices 
have  the  equations 

x'  =  1+  |V3 


and 


1  -  IV3, 


respectively;  and  the  latus 
rectum  is  2.  All  these 
results  refer  to  the  new 
axes,  of  course,  and  the 
locus  is  that  represented 
in  Fig.  121. 


Fig.  121. 


*  This  accords  with  a  result  of  the  preceding  article,  viz.  that  to  free  an 

equation  from  its  xy-term  it  is  only  necessary  to  turn  the  axes  through  a 

2  JT 
positive  acute  angle  determined  by  tan  2  6  = In  the  present  problem 

H  =  -{-2  and  A  =B  =  -  1,  hence  tan 26  =  cc  and  6  =  45°. 


296  ANALYTIC  GEOMETRY  [Ch.  XII. 

Example  2.     Given  the  equation 

4^-2  +  4:c^ +  ?/2- 18a;  +  26?/ +  64  =  0,      ...      (5) 

to  determine  the  nature  and  position  of  its  locus.  Turn  the  axes  through 
an  angle  B,  i.e.,  substitute  for  x  and  y,  respectively,  x'  cos  6  —  y'  sin  6  and 
x'  sill  6  -\-  y'  cosO;  equation  (.5)  then  becomes. 

x'2(4  cos2  0  +  sin2  ^  +  4  sin  0  cos  0) 

+  x'2j'(  -  8  cos  ^  sin  ^  +  2  cos  ^  sin  0-4:  sin^  ^  +  4  cos'^  0) 

+  y'-^  (4  shi2  e  +  cos2^  -  4  sin  B  cos  B) 

+  a;'(-18cos6'  +  26sin(9) 

+  ?/'(18sin(9  +  26cos(9)+64  =  0,  .  .  .  (6) 

in  which  6  is  to  be  so  determined  that  the  coefficient  of  x'y'  shall  be  zero. 
On  placing  this  coefficient  equal  to  zero,  it  is  at  once  seen  that  tan  2  9=^, 
from  which  it  follows  (cf.  exercise  3,  Art.  16,  second  method)  that 

sin  2  ^  =  i  and  cos  2  ^  =  | ; 

remembering    that    cos  2  ^  =  cos^  B  —  sin^  B  =  2  cos^  ^  —  1  =  1  —  2  sin^  B, 

it  is  easily  deduced  that  sin  6  =  — -  and  cos  B  =  ^^• 

Substituting  these  values  in  equation  (6),  it  becomes 
5x'2  -  '2V5x'  +  14V5  7/'  +  64  =  0, 

which  is  the  equation  of  a  parabola  whose  vertex  is  at  the  point 

/  1  63 

W5'       14a^5 

whose  focus  is  at  the  point  (  — , •  |,  whose  axis  coincides  with  the 

negative  end  of  the  ?/'-axis,  and  whose  latus  rectum  is  — -.     All  these 

results  refer  to  the  new  axes ;  the  locus  of  the  above  equation  is  given  in 
Fig.  79,  p.  178  (Art.  108). 

EXERCISES 

1.  For  the  hyperbola  in  Fig.  121  find  the  coordinates  of  the  center 
and  of  the  foci,  and  also  the  equations  of  its  axes  and  directrices,  all 
referred  to  the  axes  OX  and  OV, 


176-177.]  EQUATION   OF  SECOND  DEGREE  297 

By  first  removing  tlie  xy-ievm,  determine  the  nature  and  position  of 
the  loci  represented  by  the  following  equations.     AJso  plot  the  curves. 

2.  ?/2  _  2  V3  x^  +  3  a;2  +  6  X  -  4  ?/  +  5  =  0. 

3.  x^  —  4:  xy  -\-  "d  y'^  —  X  —  y  =  Q. 

4.  3  :r2  +  2  xi/  +  3  ?/2  -  16  ?/  +  23  =  0. 

5.  x'^  —  2  xy  +  y'^  —  Q  X  —  Q  y  -\-  d  =  0. 

177.  Test  for  the  species  of  a  conic.  It  is  often  desirable 
to  know  the  species  of  a  conic  represented  by  a  given  equa- 
tion even  wlien  it  may  not  be  necessary  to  determine  fully 
the  position  of  the  curve.  Remembering  that  every  equa- 
tion of  the  second  degree  represents  a  conic  (Art.  175),  and 
also  that  the  three  species  of  conies  may  be  distinguished 
from  each  other  by  the  number  of  directions  in  which  lines 
meeting  the  curve  at  infinity  may  be  drawn  through  any 
given  point  (Art.  131,  Note),  it  is  easy  to  find  a  test  that 
will  enable  one  to  distinguish  at  a  glance  the  kind  of  conic 
represented  by  a  given  equation. 

Let  the  given  equation  be 

Ax^  +  2Hxy-{-By'^-\-2ax-r'lFy-\-C=0.     .     (1) 

If  this  equation  be  transformed  to  polar  coordinates,  the 
origin  being  the  pole  and  the  a;-axis  the  initial  line,  so  that 
x  =  p  cos  6  and  y  =  p  sin  ^,  it  becomes 

p2(^  cos2 6  +  2ITsmecose  +  B sin2 6) 

-{-2p(acose  +  Fsme}+C=0.   .    .     (2) 

One  value  of  p,  determined  by  this  equation,  will  be  infinite 
if  its  direction  be  such  that 

A  cos2  (9  4-2  ^sin  6^  cos  6>  +  -5  sin^  0  =  0;     [Art»  10] 

i.e.,  if  B  tan2  0-\-2  fftan  6 -\- A  =  0  ; 

I.e.,  II  tan  a  = — .      .      .      (d) 


298  ANALYTIC  GEOMETRY  [Ch.  XII. 

Equation  (3)  shows  that  tan  6  will  have 
two  imaginary  values,  if  H^  —  AB  <  0 ; 

two  real  and  coincident  values,  if  H^  —  AB  =  0  ; 
two  real  and  distinct  values,  if        H^  —  AB  >  0. 

Therefore,  there  is  no  direction,  one  direction,  or  there 
are  two  directions,  resx^ectively,  in  which  a  line  meeting 
the  curve  in  an  infinitely  distant  point  may  be  drawn 
through  the  origin,  according  as 

R^  -  AB  is  <0,   =  0,  or  >  0 ; 
and  hence, 

if  M^  —  AB  <  0,  equation  (1)  represents  an  ellipse, 

if  H^  —  AB  =  0,  equation  (1)  represents  a  parabola, 

if  ff^  —  AB  >  0,  equation  (1)  represents  an  hyperbola. 

178.  Center  of  a  conic  section.  As  already  defined  (Arts. 
Ill,  117,  120),  the  center  of  a  curve  is  a  point  such  that  all 
chords  of  the  curve  passing  through  it  are  bisected  by  it. 
It  has  also  been  shown  that  such  a  point  exists  for  the 
ellipse  and  hyperbola,  ^.e.,  that  these  are  central  conies. 

If  the  equation  of  the  conic  is  given  in  the  form 

Ax^-^2E'x^-{-Bf-{-2ax+2F?/+ 0=0,     .     (1) 

the  necessary  and  sufficient  condition  that  the  origin  is  at 
the  center,  is  (7  =  0  and  F  =  0. 

For  if  the  origin  be  at  the  center,  and  (x-^,  y^  be  any 
given  point  on  the  locus  of  equation  (1),  then  (^Xy,  -y-^ 
must  also  be  on  this  locus  (because  these  two  points  are  on 
a  straight  line  through  the  origin  and  equidistant  from  it) ; 
hence  the  coordinates  of  each  of  these  points  satisfy  equa- 
tion (1), 

i.e.,  Ax^  +  2  Hx^y^  +  By^^  +  2  (^^^^  +  2  F^i  +  (7  =  0,     .     (2) 


177-179.]  EQUATION   OF  SECOND  DEGREE  299 

and     A(-x^y-\-2R(-x^}(-ij^') 

+  B(^-y,y  +  ^a^-x,)  +  2Fi-y,)+Q=0',  (3)   • 
and  equation  (3)  may  be  written  thus  : 

Ax^  +  2  Hx^y^  +  By^^  -  2  ai^- 2Fy^-\-  C  =^.     .     (4) 
Subtracting  equation  (4)  from  equation  (2)  gives 
4^1  +  4^^1  =  0; 
i.e.,  Gx^-\-Fy^  =  Q (5) 

But  equation  (5)  is  to  be  satisfied  by  the  coordinates  x-^ 
and  y^  of  every  point  on  the  locus  of  equation  (1),  and  the 
necessary  and  sufficient  conditions  for  this  are 

a=0  and  ^=0. 

179.  Transformation  of  the  equation  of  a  conic  to  parallel 
axes   through   its   center.     Let   the   equation  of   the   given 

conic  be 

Ax^-\-2Hxy  +  By'^  +  2ax-\-2Fy+C=0,     .     (1) 

and  let  the  coordinates  of  its  center  be  a  and  yS.  Then  to 
transform  equation  (1)  to  parallel  axes  through  the  point 
(a,  /3)  it  is  only  necessary  to  substitute  in  that  equation 
x'  -\-  a  and  y^  -\-  (3  for  x  and  y.     This  substitution  gives 

A(x'  +  a)2  +  2  H(x^  +  a)(^'  +  /5)  +  B(y^  +  py 

+  2(7(2:'  +  a)+21^(^''+yS)+  (7=0; 

i.e.,     Ax'^  +  2  Hx'y'  +  By'^  +  2  x'(Aa  +  ^/3  +  6^) 

+  2y'  {Ha  ^BP^-F^  +  Aa?  +  2  Ha^  +  B^ 

+  2  (7a  +  21^/3+ C^*  =  0.  .         .         .         (2) 

Since  a  and  ^  are  the  coordinates  of  the  center  (Art.  178), 
Aa-\-H^^-  a  =  ^   and  Ha  +  BP  +  F=0',     .     (3) 

*  It  is  to  be  noted  here  that  the  new  absokite  term,  le.,  the  term  free  from 
x'  and  y'  in  equation  (2),  may  be  obtained  by  substituting  a  and  j3  for  x  and 
y  in  the  first  member  of  equation  (1). 


300  ^  ANALYTIC  GEOMETRY  [Ch.  XII. 

solving  these  equations  gives 

which  are  the  coordinates*  of  the  center  of  the  locus  of 
equation  (1). 

The  constant  term  in  equation  (2)  is, 

=  a{Aa-\-Hp-\-  a)  + ^(ffa  +  B^  +  F)-{-  Ga  +  F/3 -h  0, 

=  Cra  +  Fl3  +  0,         [by  virtue  of  equations  (3)]    .     .      (5) 

=  Kf^fif )+  <^5|f)  +  ^'     tby  equation  (4)] 
ABC+2Faff-AF^-Ba''-  OH^^  A  .g. 

wherein 
A=AB0-{-2Faff-AF^-Ba^-  OR^  (cf.  Art.  67). 

Equations  (4)  show  that  the  center  of  the  locus  of  equa- 
tion (1)  is  a  definite  point,  at  a  finite  distance  from  the 
origin,  if  11^  —  AB  ^  0,  but  that  the  coordinates  of  this 
center  become  infinite  if  R^  —  AB  =  0.  Hence  (cf .  Art. 
177),  while  the  ellipse  and  hyperbola  each  have  a  definite 
finite  center,  the  parabola  may  be  regarded  as  having  a 
center  at  infinity. 

By  making  use  of  equations  (3)  and  (5),  equation  (2) 
may  be  written 

A^'^  +  2ffx'i/'  +  B^'^-jj~^=0;  .     .     (6) 

hence,  if  the  general  equation  of  an  ellipse  or  hyperbola  be 
transformed  to  parallel  axes  through  the  center  of  the  conic, 
the  coefficients   of   the   quadratic  terms  remain  unchanged, 


179-180.]  EQUATION   OF  SECOND  DEGREE  '  301 

those  of  the  first  degree  terms  vanish,  and  the  new  absolute 
term  becomes  ^ 

~  if2  _  AB' 

N'oTE.     Two  special  cases  should  be  noted : 

1)  Equation  (6)  shows  that  if  A  =  0,  the  locus  of  equation  (1)  con- 
sists of  two  straight  lines  through  the  new  origin  (cf.  Art.  67). 

2)  The  point  (a,  ^)  is  the  intersection  of  the  two  straight  lines 

Ax  ■\-  Hij  -V  G  =  0  and  Hx  +  Bij  +  F  =  0.      (cf.  eq.  (3)  above.) 

A       JT      C 

If  —=  —  =  —,  then  these'  lines  are  coincident  (Art.  38,  (ft)),  and  the 

coordinates  a  and  /5  become  indeterminate.  In  this  case,  it  may- 
be shown  that  A  =  0 ;  that  the  locus  of  equation  (1)  consists  of  two 
lines  parallel  to,  on  opposite  sides  of,  and  equidistant  from,  the  line 
Ax  -\-  Hy  +  G  =  0 ;  hence  any  point  of  the  latter  line  may  be  considered 
as  a  center,  since  chords  drawn  through  such  a  point  are  bisected  by  it, 
i.e.,  the  curve  has  a  line  of  centers.  Again,  since  W^  —  AB  =  0,  this 
locus  may  be  considered  a  special  form  of  a  parabola. 

180.  The  invariants  A  +  B  and  S^  -  AB.  In  Art.  175  it 
was  shown  tliat  a  transformation  of  coordinates  by  rotating 
the  axes  through  an  angle  6  changes  the  coefficients  of  the 
equation 

Ax^  +  2Rxi/  +  Bf-\-2ax  +  2F^-\-O=0,      .     (1) 

with  the  exception  of  the  constant  term.  It  is  true,  how- 
ever, that  certain  functions  of  these  coefficients  are  not 
changed  by  this  transformation,  e.g.,  the  sum  A -\-  B  of  the 
coefficients  of  the  x^  and  y^  terms  is  the  same  after  trans- 
formation as  before.     If  the  transformed  equation  be  written 

A'x^  +  2  ff'xT/  -f-  B'f  +  2a'x  +  2F'y+  Q=0,   .  (2) 

wherein,  as  in  Art.  175, 

A'  =  A  cos2  e  +  2  H^in  6  cos  6 -\- B  sin^  (9,      .  (3) 

5^  =  ^sin2  6>-2^sin6'cos(9  +  ^cos2^,      .  (4) 

and         2ir'  =  2^cos2  6'-(^-^)sin2  6>,       .     .     .  (5) 


302  ANALYTIC  GEOMETRY  [Ch.  Xll. 

then  the  addition  of  equations  (3)  and  (4) 
gives         A'  +  B'  =  A^B  (since  sin^^  +  cos2(9  =  1).     .      (6) 
Again,      .4' -  ^' =  2^sin  2  6" +  (^  -  ^)  cos2  (9  .     .     (7) 
hence 

^A'  -  B'y  +  4:  R'^  =  \(A  -  By  -\-  -^  R^l  (sin2  2  ^  +  cos2  2  6), 
=  (A-By  +  -^II^        ...        (8) 
i.e.,   A''^ -2A'B'  +  B'^-]-4:R"^  =  A^-2AB  +  J^  +  4:R^. 

But  by  (6), 

A'^  +  2  A'B'  +B'^  =  A^  +  2AB-\-B^; 

hence,  by  subtraction, 

H"^-A'B'  =  S'2-AB,       ...       (9) 

and  the  function  R^  —  AB  is  also  unchanged  by  the  trans- 
formation of  coordinates,  through  the  angle  6.  Moreover, 
if  a  transformation  of  coordinates  to  a  new  origin  be  per- 
formed as  in  Art.  179,  A,  B,  and  R  are  not  changed, 
nor,  therefore,  the  functions  A  +  B  and  R^  —  AB.  Such 
functions  of  the  coefficients,  which  do  not  vary  when  the 
transformations  of  Arts.  175  and  179  are  performed,  are 
called  invariants  of  the  equation  for  those  transformations. 
If,  as  in  Art.  175,  6  be  chosen  so  that 

tan20  =  ^2^,  .       .       .       (10) 

then  R'  =  0,  and  equation  (8)  becomes 

-A'B'  =  R'-AB,      ...       (11) 

2  R 
Again,  from  eq.  (10),   sin  2  6 

and  cos  2  6  = 


V{A  -By +  4.  R' 
A-B 

-y(A-By+4:R'' 


hence,  equation  (8),     A' -  B' =    ^^    .       .        .        .       (12) 

sin  2  6 

Since   sin   2  ^  is  positive  (Art.  175),  therefore  the  sign  of 
A'  —  B'  is  the  same  as  the  sign  of  R, 


180-181.] 


EQUATION   OF  SECOND  DEGREE 


308 


These  results  are  useful  in  reducing  an  equation  of  a  conic 
to  its  simplest  standard  form,  as  will  be  illustrated  in  the 
following  article. 

181.  To  reduce  to  its  simplest  standard  form  the  general  equation 
of  a  conic,  a.  Central  conic.  The  result  of  Art.  180  enables  one  to 
reduce  to  its  simplest  form  a  given  equation  of  the  second  degree,  in 
which  H^  —  AB^Oy  much  more  easily  than  by  the  method  of  Art.  175. 
If  the  equation  of  the  conic, 

Ax^  +  2  Hxy  -i-By^  +  2Gx-\-2Fy  +  C  =  0,  (1) 

be  first  transformed  to  the  center  of  the  curve  as  origin,  the  resulting 
equation  becomes  (Art.  179) 

Ax'^  +  2  Hxy  -r  By'^  +  C  =  0.        .  (2) 

If  equation  (2)  be  now 
transformed  to  axes 
O'Z"  and  O'F',  making 
the  angle  6  with  O'X' 
and  O'Y'y  respectively, 
such  that 

A-B 

it  will  become  (Art.  175) 
J['x2-f5y+C"=0,  (3) 
wherein  the  new  coeffi- 
cients are  easily  deter- 
mined by  the  relations 

C'=  Ga  +  F/3+  C 
A 

h-2-ab' 

(Art.  179), 
A'  +  B'=A  +B, 
and  -A'B'  =  H^-AB 
(Art.  180). 

Example.     Suppose  the  given  equation  to  be 

3x2 +  2x?/  + 3?/2- 16y +  20  =  0,       ^       , 
in  which     A  =  3,  H  =  1,  B  ==  ^,   G  =  0,  F=  -8,  and  C  =  20. 
Then  H^  —  AB  =  —  8,  and  the  locus  is  an  ellipse. 


Fig.  122. 


(4) 


804  ANALYTIC   GEOMETRY  [Ch.  XII. 

The  coordinates  of  the  center  are  a=:  —  1,  ^8  =  3. 
Therefore,     C'  =  &'a  +  F^+ C= -4;  .4'  +  5'  =  6,   -A'B'=-^; 
and,  since  A'  is  larger  than  B',  H  being  positive  (Art.  180), 
hence  ^'  =4,  5'  =  2; 

while  tan  2  ^  =  co  ,  and  therefore  ^  =  45°.     The  transformed  equation  is 

therefore 

4x2  +  2  2/2-4  =  0, 

O  9 


I.e. 


Y  +  |  =  i,  ...  (5) 


when  referred  to  the  axes  0'X'\  0'  Y" ;  and  the  locus  is  approximately  as 
given  in  Fig.  122. 

h.  Non-central  conic.  If  H'^  —  AB  =  0,  the  relations  of  equations  (6) 
and  (11),  Art.  180,  may  still  be  used  to  simplify  the  reduction  of  equa- 
tion (1)  to  the  standard  form  for  the  equation  of  a  parabola,  if,  as  in 
Art.  176,  the  xy-tex\x\.  be  removed  first.  In  this  case,  however,  a  better 
method  of  reduction  is  as  follows  : 

Since  the  first  three  terms  of  equation  (1)  form  a  perfect  square,  that 
equation  may  be  written 

(VTa:  +  V:B .?/)2  +  2  Cx  +  2  F^  +  C  =  0    .      .       .       (6) 

wherein  the  sign  of  the  V5  is  the  same  as  that  of  H. 

Equation  (2)  may  now  be  transformed  to  new  axes  OX'  and  0Y\ 
which  are  so  chosen  that  the  equation  of  OX'  referred  to  the  given  axes 
shall  be 

^    VAx+VBy  =  0] 

hence,  if  6  be  the  angle  between  OX  and  OX',  then 

tan  6  = ^,  whence  sin  0  =     ~  and  cos  6  =  — ^:::2z:^r    •     (J^ 

VB  '  VA  +B  VA  -{-  B 

Equation  (7)  shows  that  6  is  negative  (if  the  positive  value  of  y/A  +  B 
be  used),  and  acute  or  obtuse  according  as  V!B  is  positive  or  negative. 
The  formulas  for  transforming  to  the  new  axes  are  (cf .  Art.  72) 

x'  -\ y'  and  y  =  — ^333:::;  x'   -\ y  .      .     (0) 


y/A+B         V.4  +B  y/A+B  -J  A  +  B 

Substituting  these  values  for  x  and  y  in  equation  (6),  it  becomes 

(4  +  B)/^  +  2«^^^ZVa^,  +  2^^4±^2/'  +  C  =  0.    .    (9) 
^A+B  y/A+B 


181. J 


EQUATION  OF  SECOND  DEGREE 


305 


By  dividing  equation  (9)  by  (A  +  B),  completing  the  square  of  the 
y'-ierms,  and  transposing,  it  may  be  written  in  the  form 


_GVZj-FV^)  2 


(A+B)^ 


=  -2 


GVB  -  FVA  ( 


^,  ^  {GVA  +  Fy/B)^  -  C(A  ^-BYl 
^  2(A  +B)^(GVB-FVa)     ^ 


(10) 


(A  +  5)2 

Comparing  equation  (10)  with  equation  [42]  (Art.  106),  it  is  seen  that 
the  length  of  the  latus  rectum,  as  well  as  the  coordinates  of  the  vertex 
and  focus  (with  reference  to  the  axes  OX'  and  OY'),  and  other  impor- 
tant facts,  may  be  read  directly  from  the  equation. 

The  advantage  of  equation  (10),  over  that  resulting  from  the  reduction 
of  Ex.  2,  Art.  176,  is  that,  in  connection  with  equation  (7),  it  gives  all  the 
facts  necessary  for  the  immediate  location  of  the  curve,  and  gives  those 
facts  in  terms  of  the  coefficients  of  the  original  equation. 

Example.  Let  it  be  required  to  determine  the  position  and  parameter 
of  the  parabola  represented  by  the  equation 

9x^-2ixy  +  16  2/2  -18 a: -101 2/ +  19  =  0. 

The  given  equation  may  be  writ- 
ten as 

(3  a;  -  4  ?/)2  -  18a;  -  101 2/  +  19  =  0. 
If  the  line  Sx  —  4:y  =  0he  chosen 
as  ar'-axis,  then  tan  ^  =  f ,  whence 
sin^  =  — I,  and  cos  6  =  —  ^.  The 
formulas  of  transformation  then 
are : 

-4a;'  +  3^'       ,  3a;'  +  42/' 

x= ■ — ^  and  y= ■ — ^. 

5  5 

Substituting  these  values  in  equa- 
tion (1),  it  becomes 

25y'^-h70y'  =  -75x'  -19; 

this  equation  may  be  written 

which  shows  that  the  latus  rectum  is  3,  and  the  coordinates  of  the  vertex 
and  focus  (with  reference  to  the  new  axes)  are,  respectively,  f,  — -^  and 
—  ^,  —  i-  It  also  shows  that  the  axis  of  the  curve  is  parallel  to  the 
negative  end  of  the  x'-axis. 

Recalling  the  remark  about  the  angle  0  determined  by  equations  (7) 
above,  it  is  seen  that  the  geometric  representation  of  the  above  equation 
is  shown  in  Fig.  123. 


Fig.  123. 


TAN.   AN.    GEOM. 


20 


306  ANALYTIC  GEOMETRY  [Ch.  XII. 

182.  Summary.  It  lias  been  shown  in  the  preceding 
articles  that  every  equation  of  the  second  degree  in  two 
variables  represents  a  conic  section,  whether  the  axes  are 
oblique  or  rectangular  ;  and  that  its  species  and  position 
depend  upon  the  values  of  the  coefficients  of  the  equation. 
The  various  criteria  of  the  nature  of  the  conic  represented 
by  such  an  equation,  in  rectangular  coordinates,  appear  in 
the  following  table  : 

The  General  Equation  of  the  Second  Degree 

Ax^  +  2  Rxi/  +  Bi/  -\-  2  ax  -\-  2Fy  +  C  =  ^ 
A  =  ABC  +  2  Fas-  AF'^  -  BG^  -  HG^ 

I.     E'^-  AB<  0.     The  ellipse. 

(1)  if  A  =  B,  and  ir=0,  a  circle. 

(2)  if  A  is  +,  imaginary. 

(3)  if  A  is  — ,  real. 

(4)  if  A  is  0,  a  pair  of  imaginary  straight  lines; 

or,  a  point. 

II.     B:^-AB  =  0.     The  parabola. 

(1)  if  ^  is  + ,  principal  axis  is  the  new  ^-axis. 

(2)  if  ^  is  — ,  principal  axis  is  the  new  a;-axis. 

(3)  if  A  is  0,  pair  of  parallel  straight  lines,  which 

are  real  and  different,  real  and  coincident, 
or  imaginary,  according  as  Cr^  —  AC  >, 
=  ,  or  <  0. 

III.     R^-  AB>  0.     The  hyperbola. 

(1)  if  A=—  B^  a  rectangular  hyperbola. 

(2)  if  A  is  +,  principal  axis  is  the  new  ?/-axis. 

(3)  if  A  is  — ,  principal  axis  is  the  new  ic-axis. 

(4)  if  A  is  0,  a  pair  of  real  intersecting  straight 

lines. 


182-183.]  EQUATION   OF  SECOND  DEGREE  307 

JSToTE.  The  above  results  have  not  all  been  shown,  but  are  easily 
deduced  from  the  work  already  given.  Thus  the  locus  of  equation  (3), 
Art.  181,  if  an  ellipse,  is  imaginary  if  C"  is  —  ;  but,  by  equation  (6),  Art. 
179,  C  is  —  if  A  is  +  ;  hence  the  test  I  (2),  given  above.  And  so  for 
the  other  tests,  which  the  student  should  verify.  The  angle  0  which 
the  new  axes  make  with  the  old,  respectively,  is  chosen  as  in  Art.  175, 
2  0  being  taken  always  positive  and  not  greater  than  180°. 

183.   The  equation  of  a  conic  through  given  points.     The 

general  equation  of  a  conic  may  be  written 

Ax'^-^2ffxi/-{-Bf  +  2ax-\-2F?/+O=0,    .     (1) 

and  contains  five  parameters,  the  five  ratios  between  the 
coefficients  A^  S^  B^  (x,  F^  C.  Since  five  equations,  or  con- 
ditions, will  determine  those  parameters,  in  general  five 
points  will  determine  a  conic.  That  is,  in  general,  a  conic 
may  he  made  to  pass  through  five^  and  only  five^  given 
points. 

If,  however,  the  conic  is  to  be  a  parabola,  one  equation  is 
given  ;  viz.  H'^  —  AB  =  0,  hence  only  four  additional  con- 
ditions are  needed.  In  general,  a  parabola  may  he  made  to 
pass  through  four  points,  only. 

A  circle  has  two  conditions  given,  viz.  A  =  B,  11=  0; 
therefore,  in  general,  a  circle  may  he  made  to  pass  through 
three  points,  only. 

A  pair  of  straight  lines  has  one  condition  given,  A  =  0  ; 
therefore,  in  general,  a  pair  of  straight  lines  may  he  made 
to  pass  through  four  points,  only. 

The  method  to  be  followed  in  obtaining  the  equation  of 
the  required  conic  has  been  used  in  Art.  80,  and  may  be 
indicated  for  finding  the  equation  of  the  parabola  through 
four  given  points. 

Pi  =(2^1,  y^),  A  =(^2^  ^2)'  A  =(•'^3'  ^3)'  and  P^=(x^.  y^. 
The  equation  must  be  of  the  form  (1), 


308  ANALYTIC  GEOMETRY  [Ch.  XII.  183. 

therefore,  Ax^^  +  2  Rx^^^  +  Bi/^^  +  2  ax^-\-2  Fy^  +  C=  0, 
Ax^^  +  2Hx^^^^-By^  +  2  ax^-\-2Fy^-^  C=0, 
Ax^ + 2  ^2:3^3  +  By^^  +  2ax^  +  2Fy^+C=^, 
Ax^^2Hx^^-\-By^^  +  2ax^  +  2Fy^-\-O=^0', 

also,  H^-AB  =  0. 

The   required  ratios  between  the  coefficients  of  equation 
(1)  may  be  found  from  these  equations. 

EXAMPLES    ON    CHAPTER    XII 

Without  transforming  the  equations  to  other  axes,  find  the  center 
or  the  vertex,  the  axes,  and  the  nature  of  the  following  conies : 

1.  a:2  +  5  a:?/  +  1/2  +  8  a;  -  20  ?/  +  15  =  0  ; 

2.  (x  -yy^  +  2x  -y  =  1] 

3.  3x2+2?/2-2a:+.?/-l=0; 

4.  3a;2  -  8 a;?/  -  3 .?/2  +  a;  +  17  ?/  -  10  =  0; 

5.  4  a;2  —  4  a:?/  +  ?/2  +  4  aa:  —  2  a?/  =  0 ; 

6.  5a:2  +  2a;2/  +  5/  =  0; 

7.  3x2  +  3?/2  +  11  a;  -  5^  +  7  =  0  ; 

8.  a;2  +  2a;y-y2  +  8a;  +  4?/-8  =  0; 

9.  2/2  —  a:?/  —  6  x2  +  1/  —  3  a;  =  0  ; 

10.  y'^  —  xy  —  5  X  +  5  y  =  0. 

Trace  the  following  conies  : 

11.  3a;2  +  2a:?/  + 3?/2- 16?/ +  23  =  0; 

12.  4  a:2  +  9  ?/2  +  8  a;  +  36  ?/  +  4  =  0  ; 

13.  3  a;2  _  8  ?/2  +  8  a:?/  -  10  ?/  +  6  a;  +  5  =  0  ; 

14.  (x  -  y)(x  -y  -6)  +9  =0. 

15.  What  conic  is  determined  by  the  points  (0,  3),   (1,  0),  (2,  1), 
(-1,-3),  and  (3,-3)? 

16.  Find   the  equation  of   the  parabola  through  the  points  (3,  2), 
(1,  I),  (-6,  8),  and  ("2,  |). 

17.  Find  the  equation  of  the  conic  through  the  points  (9,  2),  (6,  3), 
(3,  2),  (1,  -2),  (2,  1). 


CHAPTER   XIII 
HIGHER  PLANE  CURVES 

184.  Definitions.  A  curve,  in  Cartesian  coordinates,  whose 
equation  is  reducible  to  a  finite  number  of  terms,  each  involv- 
ing only  positive  integer  powers  of  the  coordinates,  is  called 
an  algebraic  curve ;  all  other  curves  are  called  transcendental 
curves. 

Algebraic  curves  the  degree  of  whose  equations  exceeds 
two,  and  all  transcendental  curves,  are  (if  they  lie  wholly  in 
a  plane)  called  higher  plane  curves.  On  account  of  their 
great  historical  interest,  and  because  of  their  frequent  use 
in  the  Calculus,  a  few  of  these  curves  will  be  examined  in 
the  present  chapter. 

I.   ALGEBEAIC   CURVES 

185.  The  cissoid  of  Diodes.*  The  cissoid  may  be  defined 
as  follows  :  let   OFAK  be  a  fixed  circle  of  radius  a,  OA  a 

*  This  curve  was  invented,  by  a  Greek  mathematician  named  Diodes,  for 
the  purpose  of  solving  the  celebrated  problem  of  the  insertion  of  tv^o  mean 
proportionals  between  two  given  straight  lines.  The  solution  of  this  problem 
carries  with  it  the  solution  of  the  even  more  famous  Delian  problem  of  con- 
structing a  cube  whose  volume  shall  be  equal  to  two  times  the  volume  of  a 
given  cube.  For,  let  a  be  the  edge  of  the  given  cube ;  construct  the  two 
mean  proportionals  x  and  y  between  a  and  2 a  ;  then  a  :  x  :  -.x  :y  :  :y  -.'la, 
whence  x^  =  2-  a^,  i.e.,  x  is  the  edge  of  the  required  cube.  If  a  =  1,  then 
X  =  \/2,  hence  the  insertion  of  two  mean  proportionals  enables  one  to  con- 
struct a  line  equal  to  the  cube  root  of  2.  The  cissoid  may  also  be  employed 
to  construct  a  line  equal  to  the  cube  root  of  any  given  number  (see  Klein, 
Elementargeometrie,  S.  35,  or  the  English  translation  by  Professors  Beman 
and  Smith). 

It  is  not  positively  known  just  when  Diodes  lived  ;  it  is  very  probable, 
however,  that  it  was  in  the  last  half  of  the  second  century  b.c. 

309 


310 


ANALYTIC  GEOMETRY 


[Ch.  XIII. 


diameter,  AT  a,  tangent ;  draw  any  line  as  OQS  through  0, 
meeting  the  circle  in  Q  and  the  tangent  in  S,  and  on  this 
line  lay  off  the  distance  OP  =  QS :  the  locus  of  the  point 
P,  as  the  line  OS  revolves  about  0,  is  the  cissoid.  * 

From  this  definition,  the  equation  of  the  cissoid,  referred 
to  the   rectangular   axes    OX  and    OT",  is  readily  derived. 

Let  the  coordinates  of  P  be  a; 
and  ?/,  and  let  O  be  the  center 
of  the  circle  so  that 

00=  0A=  OK=a. 

Since   triangles   031P   and 
ONQ  are  similar, 

,'.MPi  OMi'.NQ:  (9iV;.(l) 

and  since  OP  =  QS^  therefore 

NA  =  OM  =  X  ;  moreover, 

WQ^=  0]Sr-]VA=(2a-x}x. 

Substituting  these  values  in 
equation  (1)  gives 


Fig.  124 


y  \x  '.  \  V(2  a  —  x^x  :  (2  a  —  x^, 


whence 


f  = 


x'- 


2a  —  x^ 


(2) 
(3) 


which  is  the  required  rectangular  equation  of  the  cissoid. 

The  definition  of  the  cissoid,  as  well  as  the  equation  just 
derived,  shows  that  the  curve  is  symmetric  with  regard  to 


*  Diodes  named  his  curve  ' '  cissoid ' '  (from  a  Greek  word  meaning 
"  ivy,"  because  of  its  resemblance  to  a  vine  climbing  upwards.  The  name 
"cissoid'"'  is  sometimes,  though  rarely,  applied  to  other  curves  which  are 
generated  as  stated  in  the  definition  given  above,  except  that  some  other 
basic  curve  is  employed  instead  of  a  circle.  For  other,  but  equivalent,  defini- 
tions of  the  cissoid  see  Note  3,  below. 


185.]  HIGHER  PLANE  CURVES  311 

the  a;-axis ;  that  it  lies  wholly  between  the  «/-axis  and  the 
line  x  —  2a\  that  it  passes  through  the  extremities  F  and  K 
of  the  diameter  perpendicular  to  OA  ;  and  that  it  has  two 
infinite  branches  to  each  of  which  the  line  x  =  2a  is  an 
asymptote. 

N'oTE  1.  The  polar  equation  of  the  cissoid  referred  to  the  initial  line 
OX,  and  pole  0,  is  also  easily  found.  Let  the  polar  coordinates  of  P  be 
p  and  6]  then, 

p  =  OP  =  QS  =  OS  -  OQ,  .  .  .  (4) 

but  OS  =  2a  sec  6,  and  OQ  =  2  a  cos 0, 

p  =  2a  sec  0  —  2a  cos  0  =  2  a  (sec  $  —  cos  6), 

i.e.,  p  =  2a  tan 6  sin  0,  .  .  .  ,  .  (5) 

which  is  the  polar  equation  sought. 

!N"oTE  2.  To  ''duplicate  the  cube  "  by  means  of  the  cissoid,*  extend 
CK  to  H,  making  HK  =  CK  =  a,  draw  the  line  HA  cutting  the  cissoid 
in  J,  and  draw  the  ordinate  EJ.  Since  CH  =  2  CA ,  therefore  EJ  =  2  EA  ; 
but  from  equation  (3), 

^2      OE^      OE^  _ 


EA       ^EJ' 
.-.     EJ^  =  2  0E\         ...  (6) 

Xow  let  111  be  the  edge  of  any  given  cube,  and  let  it  be  required  to 
construct  a  line  n  such  that  the  cube  on  n  shall  be  equal  to  the  double  of 
the  cube  on  m.     Construct  n  so  that 

OE  :EJ:'.m:n', 

then  OE   :  EJ    =  m^  :  n% 

and,  since  EJ^  =  2  .  OE^,  therefore  n^  =  2  m^. 

XoTE  3.  The  cissoid  may  also  be  defined  in  either  of  the  following 
ways :  (1)  as  the  locus  of  the  point  (P)  in  which  the  chord  OQS  inter- 
sects that  ordinate  {ML)  of  the  circle  which  is  equal  to  iVQ;  and  (2)  as 
the  locus  of  the  foot  of  the  perpendicular  let  fall  from  the  vertex  of  a 
parabola  upon  a  tangent.  The  derivation  of  the  equation  of  the  curve 
based  upon  these  definitions  is  left  as  an  exercise  for  the  student. 

*  To  insert  two  mean  proportionals  between  two  given  lines  by  means  of 
the  cissoid.     See  Cantor,  Geschichte  der  Mathematik,  Bd.  I.,  S.  339. 


312 


ANALYTIC  GEOMETRY 


[Ch.  XIII. 


For  Newton's  method  of  drawing  the  cissoid  by  continuous  motion, 
see  Salmon's  Higher  Plane  Curves,  p.  183,  or  Larduer's  Algebraic 
Geometry,  p.  196. 

186.   The  conchoid  of  Nicomedes.*     The  conchoid  may  Idc 

defined  as  follows  :  Let  PRP'  ^  be  a  -fixed  circle  of  radius 
a  whose  center  S  moves  along  a  fixed  straight  line  OX ;  let 
LK  be  a  straight  line  drawn  through  a  fixed  point  A  and 
the  center  S  of  this  moving  circle,  and  let  P  and  P'  be  the 
intersections  of  this  line  and  the  circle ;  then  the  locus 
traced  by  P  (and  by  P')  as  S  moves  along  OX  is  a  conchoid. 

Y 


Fig.  125 

This  definition  may  also  be  stated  thus  :  If  ^  is  a  fixed 
point,  OX  a  fixed  line,  and  S  the  point  in  which  OX  is 
intersected  by  a  line  LK  revolving  about  A^  then  the  locus 
of  a  point  P  on  LK^  so  taken  that  SP  is  always  equal  to  a 
given  constant  a^  is  a  conchoid. 

The  fixed  point  A  is  called  the  pole,  the  constant  parameter 
a  the  modulus,  and  the  fixed  line  OX  the  directrix  of  the 
conchoid. 

*  The  conchoid  was  invented  by  a  Greek  mathematician  named  Nicomedes, 
probably  in  the  second  century  b.c.  Like  the  cissoid,  it  was  invented  for  the 
purpose  of  solving  the  famous  problem  of  the  "  duplication  of  the  cube";  it 
is,  however,  easily  applied  to  the  solution  of  the  related,  and  no  less  famous, 
problem  of  the  trisection  of  a  given  angle  (see  Note  3,  below). 


186. J  HIGHER  PLANE  CURVES  313 

To  derive  the  rectangular  equation  of  the  conchoid  draw 

AOY  perpendicular,  and  J.ir  parallel,  to  OX,  and  let  OA=c; 

let  P=(x,  y)  be  any  position  of  the  generating  point,  and 

draw  the  ordinate  HMP ;  then,  from  the  similar  triangles 

ARP  and  SMP, 

AH,  HP'.  iSM.MP, 

i.e.,  x:  y  -{-  e  '.  :  Va^  —  'if",  y  \ 

[since  SM  =  ^SP  -  MP  =^a^  -  ^2], 
whence  a^y  =(jj  -\-  cy^(cfi  —  ?/2), 

which  is  the  equation  sought. 

The  definition  of  the  conchoid,  as  well  as  the  equation  just 
derived,  shows  that  the  curve  is  symmetric  with  regard  to 
the  ?/-axis  ;  that  it  lies  wholly  between  the  two  lines  y  =  ci 
and  y  =  —  a\  and  that  it  has  four  infinite  branches  to  each 
of  which  the  a;- axis  is  an  asymptote.* 

Note  1.   The  polar  equation  of  the  conchoid.     Let  A  be  the  pole,  A  Y 

the  initial  line,' and  F  =  (p,  0)  (or  P')  any  position  of  the  generating 

point;  then 

p  =  AP  =  AS  ±  SP  =  OA-secO  ±  SPy 

i.e.,  p  =  c  seed  ±a, 

which  is  the  desired  equation. 

Note  2.  The  conchoid  may  also  be  readily  constructed  by  continuous 
motion  as  follows :  By  means  of  a  slot  in  a  ruler,  fitting  over  a  pin  at  A, 
the  motion  of  the  line  LK  is  properly  controlled ;  if  now  a  guide  pin  at 
S,  and  a  tracing  point  at  P,  be  attached  to  this  ruler,  then  the  point  P 
will  trace  out  the  conchoid  when  the  guide  point  5  is  moved  along  the 
line  OX. 

Note  3.  By  means  of  a  conchoid,  any  given  angle  may  be  trisected.f 
Let  ABC  be  any  angle,  on  one  side  (BA)  take  any  distance,  as  BH,  and 

*  It  is  evident  that,  ii  AO  <  OB,  i.e.,  H  c<  a,  the  curve  has  an  oval  below 
A  as  shown  in  Fig.  2  ;  if  c  =  a,  this  oval  closes  up  to  a  point ;  and  if  c  >  a, 
both  parts  of  the  curve  lie  wholly  above  A. 

t  For  the  insertion  of  two  mean  proportionals  between  two  given  lines  by 
means  of  the  conchoid,  see  Cantor,  Geschichte  der  Mathematik,  Bd.  I., 
S.  336. 


314 


ANALYTIC  GEOMETRY 


[Ch.  XIII. 


draw  OHX  perpendicular  to  the  other  side  of  the  angle  (£C)  ;  then  lay- 
off OK  =  2  BH,  and  construct  the  conchoid  KEF  with  B  as  pole  and 
BH  =  \  OK  as  modulus,  and  OX  as  directrix.  Draw  HL  parallel  to  BC 
and  counect  5  with  L,  then  the  angle  LBC  =  \ABC'j  for,  join  D,  the 


middle  point  of  ML,  to  H,  then  ML  =  OK  =  2BH  =  2HD,  and  the 
three  angles  marked  a  are  all  equal,  as  are  also  the  two  marked  (3 ;  more- 
over, f3  =  2a,  being  the  exterior  angle  of  the  triangle  HLD,  which  proves 


that  angle  LBC  =  \ABC. 


187.  The  witch  of  Agnesi.*  The  witch  may  be  defined  as 
follows  :  Let  OKAQ  be  a  given  fixed 
circle  of  radius  a,  OA  a  diameter,  and  Q 
any  point  on  the  circle  ;  if  now  the  ordi- 
nate MQ  be  produced  to  P,  so  that 


MQ'.MPiiMA:  OA, 


(1) 


then  the  locus  of  P,  as  Q  moves  around 
the  circle,  is  the  witch.  To  derive  the 
rectangular  equation  of  the  witch,  let 
P  =  (a?,  y)  be  any  point  on  the  curve  ; 
then,  since 


MQ  ■  =  V  OM'  MA  =Vx(2a-x'). 


*  The  witch  was  invented  by  Donna  Maria  Gaetana  Agnesi  (1718-1799) 
an  Italian  lady  who  was  appointed  professor  of  mathematics  at  the  University 
of  Bologna,  in  1750. 


186-188.]  HIGHER   PLANE  CURVES  315 

substituting  in  equation  (1)  gives 

^x(2a-  X)  -.y  ::(2a-x):2a,    .     .     .      (2) 

/=H^^'  .         •         .         (3) 

whicli  is  the  equation  sought. 

The  definition  of  the  witch,  as  well  as  the  equation  just 
derived,  shows  that  the  curve  is  symmetrical  with  regard  to 
the  :?;-axis  ;  that  it  lies  wholly  between  the  «/-axis  and  the 
line  x=  2a  ',  and  that  it  has  two  infinite  branches  to  each  of 
which  the  line  a;  =  2  «  is  an  asymptote. 

188.  The  lemniscate  of  Bernouilli.*  The  lemniscate  may 
be  defined  as  follows  :  let  LTARNA' K  be  a  rectangular 
hyperbola,  0  its  center,  OX  and  01^  its  axes,  and  TE  2i  tan- 
gent to  the  curve  at  any  point  T.  Also  let  0(r  be  a  perpen- 
dicular from  the  center  upon  this  tangent,  and  let  P  be  the 
point  of  their  intersection ;  then  the  locus  of  P  as  2^  moves 
along  the  hyperbola  is  called  the  lemniscate. 

To  derive  the  rectangular  equation  of  this  curve,  let 
OA  =  a,  and  let  the  coordinates  of  T  be  x-^^  and  y^ ;  then  the 
equation  of  the  tangent  TU  is 

^1^  -  l/il/  =  ^^        •         •         •         (1) 

hence  the  equation  of   0(7,  the  perpendicular  upon  this  tan- 
gent (Art.  62),  is 

x^y  -\-  y^x  =  0.        .         .         .         (2) 

*  The  lemniscate  was  invented  by  Jacques  Bernouilli  (1654-1705),  a  noted 
Swiss  mathematician  and  professor  in  the  University  of  Basle.  It  is,  how- 
ever, only  a  special  case  of  the  Cassinian  ovals  ;  viz.,  of  the  locus  of  the  ver- 
tex of  a  triangle  whose  base  is  given  in  length  and  position,  and  the  product 
of  whose  other  two  sides  is  a  constant.  See  Salmon's  Higher  Plane  Curves, 
p.  44,  Gregory's  Examples,  or  Cramer's  Introduction  to  the  Analysis  of 
Curves. 


316 


ANALYTIC  GEOMETRY 


[Ch.  XIII. 


Regarding  equations  (1)  and  (2)  as  simultaneous,  the  x 
and  1/  involved  are  the  coordinates  of  the  point  P ;  more- 
over, since  the  point  T  =  (x^,y-^  is  on  the  hyperbola,  therefore 

.2_,/.2^^.2.  ...  (3) 


X. 


Vi  =  «' 


Eliminating  x-^  and  y^  between  equations  (1),  (2),  and  (3) 

gives 

(x^ -\- y'^y  =  a\a^  -  y'^^,      .      .       .      (4) 

which  is,  therefore,  the  equation  sought. 


The  definition  of  the  lemniscate,  as  well  as  the  equation 
just  derived,  shows  that  the  curve  is  symmetrical  with 
regard  to  both  coordinate  axes ;  that  it  lies  wholly  between 
the  two  lines  whose  equations  are  x  =  —  a  and  x  =  -{-  a  \  that 
it  passes  through  the  origin  and  the  two  points  (—  a,  0)  and 
( +  «,  0) ;  and  that  y  is  never  larger  than  x  ;  hence  the 
lemniscate  is  a  limited  closed  curve  as  represented  in  Fig.  128. 

Note  1.  The  polar  equation  of  the  lemniscate  is  easily  derived  from 
equation  (4)  if  the  x-axis  be  chosen  as  initial  line  and  the  origin  as  pole ; 


188.]  HIGHER  PLANE  CURVES  317 

for  then  x  =  p  cos  6  and  y  =  p  sin  0,  and  equation  (4)  at  once  reduces  to 

p2  =  a2(cos2^-sin2^)=a2cos2^,      ...       (5) 

which  is  therefore  the  required  polar  equation  of  the  lemniscate. 

Equation  (5)  shows  that :  when  0  =  0,  p  =±  a;  when  0 < 45°,  p  has 
two  equal  but  opposite  values,  each  of  which  is  smaller  than  a  ;  when 
0  =  45°,  p  =  0,  i.e.,  the  angle  which  the  curve  makes  with  the  initial  line 
is  45° ;  when  45° <  ^ <  135°,  p  is  imaginary ;  when  135° <0<  180°,  p  has 
two  equal  but  opposite  values,  each  of  which  is  smaller  than  a ;  and  when 
9  —  180°,  p  =±  a.  The  curve,  therefore,  consists  of  two  ovals  meeting  in 
0,  each  lying  in  the  same  angle  between  the  asymptotes  of  the  hyperbola 
as  does  the  corresponding  branch  of  that  curve,  and  these  asymptotes  are 
tangent  to  the  lemniscate  at  the  point  0. 

Note  2.    If  the  two  points  i^j  and  F  be  so  located  that 
F^O  =  OF  =  —  y/2,  and  if  S  =  (x,  y)  be  any  point  on  the  lemniscate, 


then  F^S  =  ^F^M^  +  MS""  =  ^j(^  V2  +  xY  +  y% 


and 


FS  =  ^J(^^^/2-xy+y^ 


hence  F^S  •  FS  =  ^j(^  V2  +  xY-hy^  •  -^f  |  V2'-  xY+  y^ 

=  ^|(x^  +  y^y-a^ix^-y')+J  =  |,  [by  eq.  (4)], 
i.e.,     F,S  •FS  =  ^. 

Hence  the  lemniscate  may  be  defined  as  the  locus  of  a  point  which 
moves  so  that  the  product  of  its  distances  from  two  fixed  points  is  con- 
stant, and  equal  to  the  square  of  half  the  distance  between  the  fixed 
points  (cf.  foot-note,  p.  315). 

This  definition  of  the  curve  easily  leads  to  the  equation  already 
derived ;  it  also  enables  one  to  readily  construct  the  curve  thus :  with 
F  as  center,  and  any  convenient  radius  FS,  describe  an  arc ;  then,  with 
Fj  as  center,  and  a  third  proportional  to  FS  and  OF  as  radius,  describe 
another  arc  cutting  the  first  in  S ;  this  intersection  .S  is  a  point  on  the 
locus,  and  as  many  points  as  desired  may  be  constructed  in  the  same 
way. 


318 


ANALYTIC  GEOMETRY 


[Ch.  XIII. 


189^.  The  limacon  of  Pascal.*  The  limagon  may  be  defined 
as  generated  from  a  circle  by  adding  a  constant  length  to 

each  of  the  radii  vectores 

drawn  from  a  point  on  its 

circumference  as  origin, — 

proper  account  being  taken 

of  negative  radii  vectores.f 

U.g.,  let  OLA^Whe  a  given 

— X    circle  of  radius  a,   0  any 

point  on  it,  A^A  =  k   any 

constant  ;      then    if     any 

radius    vector  as  OP^    be 

drawn  from  0,   and    PjP 

=  A^A  =  k    he  added  to 

I  it,  then  P  is  a  point  on  the  limagon ;  and  as  P^  is  made  to 

describe  a  circle,  P  will  trace  the  limagon. 

The  polar  equation  of  the  curve  is  at  once  written  down 
from  this  definition  ;  for,  if  the  diameter  OCX  be  taken  as 
initial  line,  then  the  polar  equation  of  the  circle  is 

p  =  2  a  cos  9,         .         .         .       (1) 

whence  the  polar  equation  of  the  limagon  is 

p  =  2acQse  -\-k.        .       .        .        (2) 

If  k  be  taken  equal  to  a,  the  radius  of  the  given  circle, 
this  equation  may  be  written  in  the  more  common  form 

p  =  a(l  +  2cos0}.      ...       (3) 

*  This  curve  was  invented  and  named  by  Blaise  Pascal  (1623-1662),  a 
celebrated  French  geometrician  and  philosopher.  It  is,  however,  a  special 
case  of  the  so-called  Cartesian  ovals. 

t  The  limacon  may  also  be  defined  as  the  locus  of  the  intersection  of  the 
two  lines  OP  and  CP  which  are  so  related  during  their  revolution  about  0 
and  C,  respectively,  that  the  angle  XCP  is  always  equal  to  |  times  the  angle 
XOP.     This  definition  easily  leads  to  the  polar  equation  already  derived. 


189«-1896.]  HIGHER  PLANE  CURVES  319 

The  definition  of  the  limagon,  as  well  as  the  equation  just 
derived,  shows  that  the  curve  is  symmetrical  with  regard  to 
the  initial  line,  and  that  it  has  the  form  shown  in  Fig.  129. 

Note.  The  rectangular  equation  of  the  lima9on  for  which  ^  =  a  is 
easily  derived  from  equation  (3).  Choosing  the  initial  line  and  a  perpen- 
dicular to  it  through  0  as  rectangular  axes,  so  that  x  =  p  cos  0,  and 
y  =  psin  6,  equation  (3)  becomes 

Vx-2  +  //^  =  a  +  2  g .       ...         (4) 

y/x"^  +  y'^ 
Rationalizing  equation  (4)  gives 

(x2  +  f  -2  axy  =  a%x^  +  tf),        .         ...         (5) 

which  is  the  usual  form  for  the  rectangular  equation  of  the  lima9on. 

189^.  The  cardioid.  The  cardioid  may  be  defined  as  a 
special  case  of  the  limagon ;  viz.,  it  is  a  limagon  in  which 
the  constant  k,  which  is  added  to  each  of  the  radii  vectores, 
is  taken  equal  to  the  diameter  of  the  fundamental  circle. 
If  in  the  equation  of  the  limagon  [Art.  189a,  equation  (2)] 
the  constant  k  be  taken  equal  to  2  a,  that  equation  becomes 

p  =  2a(l  +  cos  6>),       .        .         .         (1) 
which  is  the  polar  equation  of  the  cardioid. 

The  more  usual  form  in  which  the  equation  of  the  cardioid 

is  written  is 

/3  =  2a(l  —  cos^),        ...        (2) 

but  this  amounts  merely  to  turning  the  figure  through  180° 

in  its  own  plane. 

Note  1.    The  rectangular  equation  of  the 
cardioid  is  obtained  as  in  Art.  189  a.  p^ » 

It  is  (x^  +  ?/2  +  2  axy  =  a%x^  +  y^) .     (3)         /%\    ^ 
The  curve  represented  by  equations  (2)       f/^     \/\ 

and  (3)  has  the  form  shown  in  Fig.  130.         -I Mi 

The   cardioid   is   usually  defined  as   the       vv_^^  \. 

locus  traced  by  a  point  on   a   given   circle        \^  ^ 

AKAyL,  which  rolls  on  an  equal  but  fixed  >v 

circle  OMA.H.    This  definition  also  leads  to  ' 

Fig  130 
equations  (2)  and  (3)  already  derived. 


320 


ANALYTIC  GEOMETRY 


[Ch.  XIII. 


190.   The  Neilian,  or  semi-cubical,  parabola.*     This  curve 
may  be  defined  as  follows  :  let  HTASKL  be  a  given  parab- 
ola whose  equation  is 

let  TMS  be  any  double  ordinate  of 
the  curve,  TT^  a  tangent  at  the  point 
-X   T=(x^^  ?/j),  and  AQ  d^.  perpendicular 
from  the  vertex  upon  this  tangent ; 
if  QA  intersects  TS  in  P,  then  the 
locus  of  P  as  2^  moves    along  the 
parabola  is  called  a  semi-cubical  or 
Neilian  parabola. 
Its  rectangular  equation  is  derived  as  follows  :  the  equa- 
tion of  r2\is 

y^y  =  1p(x  +  x;),      .       .       .       (2) 

hence  the  equation  of  ^^  is 


Fig.  131 


y=-fy- 


(3) 


The  equation  of  TS  is 

JO  —  Ju-t»  •  •  •  I  "x  J 

If  now  equations  (3)  and  (4)  be  regarded  as  simultaneous, 

then  X  and  y  are  the  coordinates  of  the  point  P  in  which  the 

two  lines  intersect,  and  if  x-^  and  y^  be  eliminated  by  means 

of  the  equation 

y^==4,px-^,  ...  (5) 

an  equation  connecting  x  and  y  is  obtained. 

*  This  curve  is  historically  interesting,  because  it  is  the  first  one  which 
was  rectified^  i.e.,  it  is  the  first  one  the  length  of  an  arc  of  which  was 
expressed  in  rectilinear  units.  This  celebrated  rectification  was  performed, 
without  the  aid  of  the  modern  Calculus  methods,  by  William  Neil,  a  pupil  of 
Wallis  (see  Cantor,  Geschichte  der  Mathematik,  Bd.  IL,  S.  827),  in  1657,  and 
is,  for  that  reason,  called  the  Neilian  parabola.     It  is  also  called  the  semi- 

3 

cubical  parabola  because  its  equation  may  be  written  in  the  form  y  =  ax^. 


190-191.] 


HIGHER  PLANE  CURVES 


321 


Substituting  for  x^  and  y^,  in  equation  (5),  their  values  in 
terms  of  x  and  ^  as  found  from  equations  (3)  and  (4),  gives 


t.e.<, 


4^2 

f: 

=  i2 

f 

(6) 


which  is  the  equation  sought. 

This  equation  shows  that  the  curve  passes  through  the 
origin  and  is  symmetrical  with  regard  to  the  a:;-axis ;  that 
it  lies  wholly  on  the  same  side  of  the  y-axis  as  does  the 
given  parabola ;  and  that  it  has  two  infinite  branches. 


II.    TRANSCENDENTAL   CURVES* 

191.   The   cycloid. t     The   cycloid   (^OPKA)   is   the   path 
traced  by  a  point  P  on  the  circumference  of  a  circle  {HNSP^ 
Y 


*  A  few  very  common  transcendental  curves  have  already  been  examined 
in  Chapter  III ;  among  these  are  the  curve  of  sines,  the  curve  of  tangents, 
and  the  logarithmic  curve. 

t  Because  of  the  elegance  of  its  properties,  and  because  of  its  numerous 
applications  in  mechanics,  the  cycloid  is  the  most  important  of  the  transcen- 
dental curves.  It  has  the  added  historical  interest  of  being  the  second  curve 
that  was  rectified  (cf.  Art,  190,  foot-note).  Its  rectification  was  first  accom- 
plished by  Sir  Christopher  Wren  (1632-1723)  and  published  by  him  in  1673. 


TAN.  AN.   GEOM.  21 


322  ANALYTIC  GEOMETBY  [Ch.  Xm. 

which  rolls,  without  sliding,  upon  a  fixed  right  line  (OX). 
The  point  P  is  called  the  generating  point ;  the  circle  PHNS^ 
the  generating  circle ;  the  points  0  and  A^  the  vertices ;  the 
line  EK,  perpendicular  to  OA  at  its  middle  point,  the  axis ; 
and  the  line  OA^  the  base  of  the  cycloid. 

To  derive  the  rectangular  equation  of  the  cycloid  let  a  be 
the  radius  of  the  generating  circle,  and  OX  the  fixed  straight 
line  on  which  it  rolls ;  also  let  P  be  the  generating  point, 
and  let  PNS  be  any  position  of  the  generating  circle. 
Draw  the  radius  CP^  the  ordinate  MP^  the  line  PL  parallel 
to  OJl,  and  the  radius  OH  to  the  point  of  contact  of  the 
generating  circle  and  the  line  OX.  Let  OX  and  OY  (the 
perpendicular  to  it  through  0)  be  chosen  as  axes,  and  let 
e  be  the  angle  PCH. 

Then,  if  P  =(x.,  ?/), 

x=  0M=  OH-MH 

=  OH -PL 

=  aO  —  a  sin  ^,    [since  0H=  'drGPH=  a6^. 

i.e.,  X  —  a(^0  —  sin  0^.  .  .  .  (1) 

Similarly,      q/  =  a(l  —  cos  ^).  .  .  .  (2) 

Solving  equation  (2)  for  d  gives 

/I      a  —  y 

cos  6  = ^, 

a 

i.e.,  6  =  cos-^ ( ^  )  =  vers-^ (  ^  ) ; 

\    a    J  \aj 

and  substituting  this  value  of  0  in  equation  (1)  gives 

x  =  a  vers~^  (     )  ~"  V2  ay  —  y'^,     .      .      .      (3) 
which  is  the  rectangular  equation  sought. 


191-192.] 


HIGHER   PLANE  CURVES 


323 


Note  1.  It  is  usually  simpler  to  regard  equations  (1)  and  (2)  together 
as  representing  the  cycloid ;  6  is  then  the  independent  variable,  while  x 
and  y  are  both  functions  of  it. 

Note  2.  The  cycloid  belongs  to  the  kind  of  curves  called  roulettes. 
These  curves  are  generated  by  a  point  which  is  invariably  connected 
with  a  curve  which  rolls,  without  sliding,  upon  a  given  fixed  curve. 

If  both  the  rolling  and  the  fixed  curves  are  circles,  then  the  curve 
generated  is  designated  by  the  general  name  of  trochoid.  If  the  gen- 
erating point  is  on  the  circumference  of  the  rolling  circle,  and  this  circle 
rolls  on  the  outside  of  a  fixed  circle,  then  the  curve  described  is  called  an 
epicycloid  ;  but  if  it  rolls  on  the  inside  of  the  fixed  circle,  the  generated 
curve  is  called  a  hypocycloid.  The  cycloid  may  be  regarded  either  as 
an  epicycloid  or  a  hypocycloid,  for  which  the  fixed  circle  has  its  center 
at  infinity  and  an  infinite  radius. 

192.  The  hypocycloid.  Let  the  hypocycloid  APRST  •• 
be  traced  by  the  point  P  on  the  circumference  of  the  circle 
PQR,  whose  radius  is  6,  and  which  rolls  on  the  inside  of  the 

Y 

s 


FigI 133 


fixed  circle  AQE,  whose  radius  is  a.  Also  let  P=  (x,  y) 
be  any  position  of  the  generating  point.  Draw  the  line 
OOQ,  the  ordinates  EO  and  MP,  the  radius  OP,  and  the 


324  ANALYTIC   GEOMETRY  [Ch.  Xlll. 

line  KF  parallel  to  OA,  where  A  is  the  point  with  which  P 
coincided  when  in  its  initial  positiono  Let  OAX  and  OY, 
the  perpendicular  to  it  through  0,  be  chosen  as  coordinate 
axes;  also  let  the  angles  AOQ,  FO'Q  and  O'FK  be  desig- 
nated, respectively,  by  6,  6'  and  <^. 

Then     0M=  OH  +  EM=  OH -\-  KF 

=  O0'cosl9  +  P0'cos(^ 

=  Oa  cos  e  +  FO'  COS  {6'  -  6>), 

[since  <^  =  S'  -  6^ 

i.e.,  x=(a-  h)  cos6  +  b  cos  (6'  _  6>).     .     .     .     (1) 

But  since  arc  AQ  =  sltc  FQ,  therefore  a0  =  b6',  whence 

$'  =-^,  and  equation  (1)  becomes 
b 

x  =  (a-b)cose  +  b  cos^^  ~  ^^  ^.    ...     (2) 

Similarly,   ^  =(^a  —  b')  sinO  —  b  sin^^ —     ^    .     .    .     .    (3) 

Equations  (2)  and  (3)  are  together  the  equations  of  the 
hypocycloid.  A  single  equation  representing  the  same 
curve  may  be  found,  as  in  the  case  of  the  cycloid  (Art.  191), 
by  eliminating  6  between  equations  (2)  and  (3). 

Note.  If  the  radii  of  the  circles  be  commensurable,  i.e.,  if  b  equals  a 
fractional  part  of  a,  then  the  hypocycloid  will  be  a  closed  curve  ;  but  if 
these  radii  are  incommensurable,  then  the  curve  will  not  again  pass 
through  the  initial  point  A. 

In  particular,  if  0:6  =  4:1,  then  the  circumference  of  the  fixed  circle 
is  4  times  that  of  the  rolling  circle,  and  the  hypocycloid  becomes  a  closed 
curve  of  four  arches,  as  shown  in  Fig.  134.  In  this  case,  equations  (2) 
and  (3)  become,  respectively, 


192-194.] 


HIGHER   PLANE  CURVES 


325 


2 


X  =  ^a  cos  ^  +  ^ a  cos  3  ^, 
and  y  =  ^a^mS  —  \a  sin  3  0. 
But,  by  trigonometry, 
3cos^+  cos  3^=i4cos3^, 
and  3  sin  0  —  sin  3^  =  4  sin^  0, 
hence  equations  (4)  become 

X  =  a  cos^  0, 
and  y  =  «sin^  0; 

2.  2l 

whence   x^  -\-  y^ 

which  is  the  common  form  of  the 
equation  of  the  four-cusped  hypocy- 
cloid. 

SPIRALS 

193.  A  spiral  is  a  transcendental  curve  traced  by  a  point 
which.,  while  it  revolves  about  a  fixed  point  called  the  center, 
also  continually  recedes  from  this  center,  according  to  some 
definite  law. 

The  portion  of  the  spiral  generated  during  one  revolution 
of  the  tracing  point  is  called  a  spire ;  and  the  circle  whose 
radius  is  the  radius  vector  of  the  generating  point  at  the 
end  of  the  first  revolution  is  called  the  measuring  circle  of 
the  spiral.  Thus,  in  Fig.  135,  ABODE  is  the  measuring 
circle,  OQSUWA  is  the  first  spire,  and  J.l^_ZrXiV  is  the  sec- 
ond spire. 

194.  The  spiral  of  Archimedes. f  This  curve  is  traced  by 
a  point  which  moves  about  a  fixed  point  in  a  plane  in  such  a 

*  If  this  equation  be  rationalized,  it  becomes 


27  a2xV  ^  (^: 


A  —  w2>3 


2/2)3. 


Although  the  hypocycloid  is,  in  general,  a  transcendental  curve,  it  becomes 
algebraic  for  particular  values  of  the  ratio  of  the  radii  of  the  circles. 

t  This  curve  is  usually  supposed  to  have  been  discovered  by  Conan, 
though  its  principal  properties  were  investigated  by  the  geometer  whose 
name  it  bears. 


326 


ANALYTIC  GEOMETRY 


[Ch.  XIII. 


way  that  any  two  radii 
vectores  are  in  the  same 
ratio  as  are  the  angles  they 
make  with  the  initial  line.* 
From  this  definition  it 
follows  that  the  equation 
of  the  curve  is 

p  =  ke,  .  .  .  (1) 

where  A;  is  a  constant. 
This  equation  shows  that 
the  locus  passes  through  the  origin,  and  that  the  radius 
vector  becomes  larger  and  larger  without  limit  as  the  num- 
ber of  revolutions  increases  without  limit.  Moreover,  if 
(.Pv  ^i)  ^®  ^^y  point  on  the  curve,  and  if  (/O2,  ^^  +  2  tt)  be 
the  corresponding  point  on  the  next  spire,  then 

p^  =  hO-^  and  p^  —  k(6^  +  2  tt), 

whence  p^  =  p^-\- '2k7r\ 

but  2k7r  =  OA^  hence  the  distance  between  the  successive 
points  in  which  any  radius  vector  meets  the  curve  is  constant ; 
it  is  always  equal  to  the  radius  of  the  measuring  circle.  This 
follows  also  directly  from  the  definition. 

The  locus  of  equation  (1),  for  positive  values  of  6  is  rep- 
resented in  Fig.  135  ;  for  negative  values  of  6  the  locus  is 
symmetrical  with  the  part  already  drawn,  the  axis  of  sym- 
metry being  the  line  LF. 

195.  The  reciprocal  or  hyperbolic  spiral.  This  curve  is 
traced  by  a  point  which  moves  about  a  fixed  point  in  a 
plane  in  such  a  way  that  any  two  radii  vectores  are  in  the 

*  This  curve  may  also  be  defined  thus  :  It  is  the  path  traced  by  a  point 
which  moves  av^ay  from  the  center  with  uniform  linear  velocity,  while  its 
radius  vector  revolves  about  the  center  with  uniform  angular  velocity. 


194-195.]  HIGHER  PLANE  CURVES  327 

same  ratio  as  the  reciprocals  of  the  angles  which  they  form 
with  the  initial  line. 

From  this  definition  it  follows  that  the  equation  of  the 
curve  is 

p  =  -,  .  .  .  (1) 

where  ^  is  a  constant. 

This  equation  shows  that  the  curve  begins  at  infinity 
when  ^  =  0  and  winds  round  and  round  the  center,  always 
approaching  it,  but  never  quite  reaching  it  ;  i.e.,  p  =  0  only 
after  an  infinite  number  of  spires  have  been  described. 

Equation  (1)  also  shows  that  the  constant  k  is  the  cir- 
cumference of  the  measuring  circle.  For  the  radius  of  the 
measuring  circle  (Art.  193)  is  the  radius  vector  of  the  gener- 
ating point  of  the  curve  at  the  end  of  the  first  revolution, 
i.e.,  when  ^  =  2  tt  ;  but,  from  equation  (1),  this  radius  vector 

is  — ■,  and  the  circumference  of  the  circle  of  which  this  is 

27r 

the  radius  is  k. 

Again,  if  P  =  (/?,  ^)  be  any  point  on  the  locus  of  equa- 
tion (1),  then 

pe  =  k 

=  circumference  of  measuring  circle ; 

but  pO  equals  the  length  of  the  circu- 
lar arc  described  with  radius  p  and 
subtending  an  angle  0,  therefore  the 
length  of  any  circular  arc  as  MP, 
described  about  0,  with  radius  p,  and 
extending  from  the  initial  line  to 
the  curve,  is  equal  to  the  circum- 
ference  of  the  measuring   circle. 

The  locus  of  equation  (1),  for  positive  values  of  6,  is 
represented  in  Fig.  136. 


328  ANALYTIC  GEOMETRY  [Ch.  XIII. 

196.  The  paracolic  spiral.  This  curve  is  traced  by  a 
point  which  moves  around  a  fixed  point  in  a  plane  in  such 
a  way  that  the  squares  of  any  two  radii  vectores  are  in  the 
same  ratio  as  are  the  angles  which  they  form  with  the 
initial  line. 

From  this  definition  it  follows  that  the  equation  of  the 

•^•^'■^'^^  ^  =  ke,      .       .       .       (1) 

where  A;  is  a  constant. 

This  equation  shows  that  the  curve  begins  at  the  center 

when  ^  =  0,  winds  round  and  round 
this  point,  always  receding  from  it, 
the  radius  vector  becoming  infinite 
when  6  becomes  infinite,  ^.e.,  when 
-R  it  has  described  an  infinite  number 
of  spires. 

The  locus   of   equation   (1),   for 
positive  values  of  /o,  is  represented 

Fig.  137  ^j^    ^ig.    137.* 

197.  The  lituus  f  or  trumpet.  This  curve  is  traced  by  a 
point  which  moves  around  a  fixed  point  in  a  plane  in  such 
a  way  that  the  squares  of  any  two  radii  vectores  are  in  the 
same  ratio  as  the  reciprocals  of  the  angles  which  they  form 
with  the  initial  line. 

From  this  definition  it  follows  that  the  equation  of  the 

curve  is  P^  —  a^  •  •  •  C^) 

u 

where  A;  is  a  constant. 

This  equation  shows  that  the  curve  begins  at  infinity, 
when  ^  =  0,  and  winds  round  and  round  the  center,  always 

*  See  also  Eice  and  Johnson's  Differential  Calculus,  p.  307. 

t  This  curve  was  invented  and  named  by  Cotes,  who  died  in  1716. 


196-198.] 


HIGHER   PLANE  CURVES 


329 


approaching  it,  but  never  quite  reaching  it,  i.e.^  f>  =  0  only 
after  an  infinite  number  of  spires  have  been  described. 

The  locus  of  equation  (1)  is  shown  in  Fig.  138  ;   the  heavy 


Fig.  138 


line  being  the  part  of  the  locus  obtained  from  the  positive 
values  of  p,  while  the  dotted  part  belongs  to  the  negative 
values  of  p. 

Note.  The  four  spirals  just  discussed,  and  whose  forms  are  given  in 
Figs.  135  to  138,  are  all  included  under  the  more  general  case  of  the  curve 
defined  by  the  equation  _    ^^ 

if  n  =  1,  this  is  the  spiral  of  Archimedes;  if  n  =  —  1,  it  is  the  hyperbolic 
spiral ;  if  n  =  |,  it  is  the  parabolic  spiral ;  while  if  n  =  —  I,  it  is  the 
lituus. 

198.  The  logarithmic  spiral.*  This  curve  is  traced  by  a 
point  which  moves  around  a  fixed  point  in  a  plane  in  such 


*  This  curve  might  have  been  defined  by  saying  that  the  radius  vector 
increases  in  a  geometric  ratio  while  the  vectorial  angle  increases  in  an  arith- 
metic ratio.  An  important  property  of  this  curve  is  (see  McMahon  and 
Snyder's  Differential  Calculus,  Art.  120)  that  it  cuts  all  the  radii  vectores 
at  the  same  angle,  and  the  tangent  of  this  angle  is  the  modulus  of  the  system 
of  logarithms  which  the  particular  spiral  represents. 


330  ANALYTIC  GEOMETRY  [Ch.  XUI.  198. 

a  way  that  the  logarithms  of  any  two  radii  vectores  are  in 
the  same  ratio  as  are  the  angles  Avhich  these  lines  form  with 
the  initial  line. 

From  this  definition  it  follows  that  the  equation  of  the 

curve  is 

log  p  =  kd^  .  .  .         (1) 

where  Jc  is  a  constant. 

If  k  be  unity,  and  logarithms  to  the  base  a  be  employed^ 
this  equation  may  be  written  in  the  form 

p  =  a'.  .  .  .  (2) 

This  equation  shows  that  if  ^  =  —  oo,  p=  0 ;  that  p  in- 
creases from  0  to  1,  while 
6  increases  from  —  go  to 
0  ;  and  that  p  continues 
to  increase  from  1  to  go, 
while  0  increases  from  0 
to  +  GO  ;  the  curve  has, 
therefore,  an  infinite  number  of  spires. 

If  the  constant  a  equals  2,  then  p  takes  the  values  •••■!,  |-, 
1,  2,  4,  8,  •••,  when  6  is  assigned  the  values  (in  radians), 
...^  _  2,  —  1,  0,  1,  2,  3,  ••• ;  Fig.  139  represents  the  locus  of 
equation  (2),  a  being  equal  to  2,  for  values  of  6  from  —  2  tt 
to  +3.  In  this  figure  Z  FOE  =  Z  EOA  =Z  AOB  =Z  BOO 
=  ZO0B=bT.%,  and  OF  =  \,  OE  =  ^,  0A=1,  OB  =  2, 
0(7=4,  and  OB  =  S. 


Fig.  139 


PART   II 

SOLID  ANALYTIC   aLOMETRY 


CHAPTER   I 
COORDINATE  SYSTEMS.     THE   POINT 

199.  Solid  Analytic  Geometry  treats  by  analytic  methods 
problems  which  concern  figures  in  space,  and  therefore  in- 
volve three  dimensions.  It  is  evident  that  new  systems  of 
coordinates  must  be  chosen,  involving  three  variables ;  and 
that  the  analytic  work  will  therefore  be  somewhat  longer 
than  in  the  plane  geometry.  On  the  other  hand,  since  a 
plane  may  be  considered  as  a  special  case  of  a  solid  where 
one  dimension  has  the  particular  value  zero,  it  is  to  be 
expected  that  the  analytic  work  with  three  coordinate  vari- 
ables should  be  entirely  consistent  with  that  for  two  vari- 
ables ;  merely  a  simple  extension  of  the  latter.  The  student 
should  not  fail  to  notice  this  close  analogy  in  all  cases. 

In  the  present  chapter  will  be  considered  some  simple  and 
useful  sj^stems  of  coordinates  for  determining  the  position  of 
a  point  in  space,  some  elementary  problems  concerning  points, 
and  the  transformations  of  coordinates  from  one  system  to 
another.  Later  chapters  Avill  treat  briefly  of  surfaces,  par- 
ticularly of  planes  and  of  surfaces  of  the  second  order,  and 
of  the  straight  line. 

331 


332 


ANALYTIC  GEOMETRY 


[Ch.  I. 


Z 


N 


X- 


,Y' 


O 


N 


^M 


^X 


M' 
Fig.  140 


200.  Rectangular  coordinates.  Let  three  planes  be  given 
fixed  in  space  and  perpendicular  to  each  other,  —  the  coordi- 
nate planes  XOY^  YOZ^  and 
ZOX,  They  will  intersect 
by  pairs  in  three  lines,  X' X^ 
Y'Y,  and  Z'Z,  also  perpen- 
dicular to  each  other,  called 
the  coordinate  axes.  And 
these  three  lines  will  meet 
in  a  common  point  0,  called 
the  origin.  Any  three  other 
planes,  MF,  AT,  and  LF, 
parallel  respectively  to  these 
coordinate  planes,  will  intersect  in  three  lines,  A^'P,  L'P^ 
M'P^  which  will  be  parallel  respectively  to  the  axes ;  and 
these  three  lines  will  meet  in,  and  completely  determine, 
a  point  P  in  space.  The  directed  distances  A'P,  X'P,  and 
M'P  thus  determined,  i.e.^  the  perpendicular  distances  of 
the  point  P  from  the  coordinate  planes,  are  the  rectangular 
coordinates  of  the  point  P.  They  are  represented  respec- 
tively by  ic,  ?/,  and  z.     It  is  clear  that 

X  ==  N'P  =  LL'    =  NM'  =  OM; 
y  =  HP  =  MM'  =  LN'  =  ON-, 
z  =  M'P  =  AAT'  =  ML'  =  OL. 
It  is  generally  convenient,  however,  to  consider 
x=  OM,  y  =  MM',  and  z  =  M'P. 
The  point  may  be  denoted  by  the  symbol  P  =  (^x,  y,  z). 

The  axes  may  be  directed  at  pleasure  ;  it  is  usual  to  take 
the  positive  directions  as  shown  in  the  figure.  Then  the 
eight  portions,  or  octants,  into  which  space  is  divided  by  the 
coordinate  planes,  will  be  distinguished  completely  by  the 
signs  of  the  coordinates  of  points  within  them. 


200-202.]  THE  POINT  IN  SPACE  333 

If  the  chosen  coordinate  planes  were  oblique  to  each 
other,  a  set  of  oblique  coordinates  for  any  point  in  space 
miglit  be  found  in  an  entirely  analogous  way. 

Unless  otherwise  stated,  rectangular  coordinates  will  be 
used  in  the  subsequent  work. 

201.  Polar  coordinates.  A  second  method  of  lixing  the 
position  of  a  point  in  space  is  by  means  of  its  distance  and 
direction  from  a  given  fixed  point.  Let  ^ 
0  be  a  fixed  point  in  space,  called  the 
pole ;  and  let  p  be  the  distance  from 
0  to  any  other  point  P.  To  give  the 
direction  of  p,  let  OH  and  OS  be  two 
chosen  directed  perpendicular  lines 
through  0,  determining  the  plane 
ROS;  then  the  direction  of  p  will  be  Fig.  ui 
given  by  the  angle  6  from  the  plane  BOS  to  the  plane  POJ/, 
and  the  angle  (f)  from  the  line  OS  to  p.  The  point  P  is 
completely  determined  by  the  values  of  its  radius  vector  p 
and  its  vectorial  angles  0  and  <^,  and  may  be  denoted  as 
P  =  (p,  6,  (/)).  The  elements  p,  6,  cj)  are  called  the  polar 
coordinates  of  the  point  P. 

It  is  to  be  noted  that  for  convenience  the  positive  values 
of  6  and  ^  are  those  for  rotation  in  clockiuise  direction  from 
ROS  and  OS^  respectively.  And  although  a  given  set  of 
coordinates  fixes  a  single  point,  yet  any  point  may  have  eight 
sets  of  coordinates  in  a  polar  system,  if,  as  usual,  the  valuer 
of  the  angles  are  less  than  360°. 

202.  Relation  between  the  rectangular  and  polar  systems. 
If  the  axes  OR  and  OS  of  a  polar  system  coincide  with 
the  axes   OX  and    OZ,  respectively,   of  a  rectangular  sys- 


3M 


ANALYTIC  GEOMETRY 


[Ch.  I. 


and 
that  is, 


Again, 
^.e., 

also 


tern,  the  pole  and  origin  therefore 
being  coincident,  then  simple  rela- 
tions exist  between  the  two  sets  of 
coordinates  for  any  point.    For,  since 

Z  OMM'  =  90°  and  Z  OM'F=  90°, 
therefore   OM ^  OM' cone 
=  OP  sin  </>  cos  0. 
MM'  =  OM'  sin  0  =  OF  sin  </>  sin  6, 
M'F  =  OP  cos  (^  ; 
a?  =  pcos0sm<|),  1 

2/ =  P  sin  e  sin  <}),  f  .  •  .  [1] 

s  =  pcos4>.         J 

OF^  =  OW^  +  WF^  =  OM^  +  MM^  +  M^. 


p2=:Cc2  +  |/2  +  »2, 


tane=^, 


a? 


and 


cos<t>  = 


^^a?'^  +  y^  -i-z^. . 


[2] 


The    above    relations    give    formulas    for   transformation 
from  the  one  coordinate  system  to  the  other. 

203.  Direction  angles :  direction  cosines.  A  third  useful 
method  of  fixing  a  point  in  space 
is  a  combination  of  the  two 
methods  already  considered. 
The  axes  of  reference  are  chosen 
as  in  rectangular  coordinates, 
and  any  point  F  of  space  is  fixed 
by  its  distance  from  the  origin, 
called  the  radius  vector,  and  the 
angles  a,  /3,  7,  which  this  radius 


Fig,  143 


202-203.]  THE  POINT  IN  SPACE  335 

vector  makes  with  the  coordinate  axes,  respectively.  These 
angles  are  called  the  direction  angles  of  the  line  OP,  and 
their  cosines,  its  direction  cosines.  The  point  may  be  con- 
cisely denoted  as  the  point  P  =  (/o,  a,  /3,  7) . 

Simple  equations  connect  these  coordinates  with  those  of 
the  rectangular  system ;  for,  projecting  OP  upon  the  axes 
OX,  OY,  and  OZ,  respectively, 

a?  =  pCOSa,     t/  =  pCOSp,     »  =  pCOSYj       .       .       .        [3] 

and  also,  p^  =  x^  -{-  y'^  -\-  z^,  as  in  equations  [2]. 

Moreover,  the  direction  cosines  are  not  independent,  but 

are  connected  by  an  equation ;  for,  by  combining  the  above 

equations, 

^  =  p2  qqq2  a-i-  p^  cos^  /3  +  p^  COS^  y^ 

i.e.,  cos'^a  + cos^|3  +  cos''^7  =  1.       .       .       .       [4] 

Such  a  relation  was  to  have  been  expected,  since  only 
three  magnitudes  are  necessary  to  determine  the  position  of 
a  point,  and  therefore  the  four  numbers  /o,  «,  yS,  7  could  not 
be  independent. 

Any  three  numbers,  a,  b,  c,  are  proportional  to  the  direc- 
tion cosines  of  some  line ;  because  if  these  numbers  are  con- 
sidered as  the  coordinates  of  a  point,  then  the  direction 
cosines  of  the  radius  vector  of  that  point  are,  by  eq.  [3], 

cos   a=—    ^  ,  COS  3= ^  ,  COS  7= .     [51 

Va2  +  ft2  +  c2  Va2  +  52  +  c2  Va2^_2,2_,.c2 

These  direction  cosines  are  proportional  to  a,  b,  c;  and  are 
found  by  dividing  a,  b,  c,  respectively,  by  the  same  constant, 

Va2  +  52  +  c2. 

Direction  cosines  are  useful  in  giving  the  direction  of  any 
line  in  space.  The  direction  of  any  line  is  the  same  as 
that  of  a  parallel  line  through  the  origin,  therefore  the  direc- 
tion of  a  line  may  be  given  by  the  direction  angles  of  some 


886 


ANALYTIC   GEOMETRY 


[Ch.  I. 


point  wliose  radius  vector  is  parallel  to  the  line.  Sometimes, 
as  an  equivalent  conception,  it  is  convenient  to  consider  the 
direction  angles  as  those  formed  by  the  line  with  three  lines 
which  pass  through  some  point  of  the  given  line,  and  are 
parallel,  respectively,  to  the  coordinate  axes. 

204.   Distance  and  direction  from  one  point  to  another  ;  rec- 
tangular coordinates.    A  few  elementary  problems  concerning 

points  can  now  be  easily  solved  ; 
for  example,  the  problem  of  find- 
ing the  distance  between  two 
points.  Let  OX,  OY,  OZ  be 
a  set    of  rectangular  axes,  and 

be  two  given  points.  Then  the 
planes  through  P^  and  P^^  paral- 
lel, respectively,  to  the  coordi- 
nate planes,  form  a  rectangular 

parallelopiped,  of  which  the  required   distance   P^P^   is   a 

diagonal.     From  the  figure, 

since  Z  P^QP^  =  90°  and  Z  M^RM^  =  90°, 

therefore  P^  =  I^Q"  +  W?  =  WW  +  ~Q^i^ 

=  MlR^  -f-  BM^  -{-  QP^ 

=  (^2  -  ^l)^  +  (^2  -  ^l)^  +  C^2  -  ^lY' 

That  is,  if  d  be  the  required  distance, 


Fig.  141 


d=  ^(0^2  -  a?i)^  +  (2/2  -  2/i)-  +  ^z,2-  zi)^ 


[6] 


Moreover,  since  the  direction  of  the  line  PjPg  ^^  given  by 
the  angles  a,  /3,  7,  which  it  makes,  respectively,  with  the  lines 
P^X^  ^lY'^  and  P^Z',  drawn  through  Pj  parallel  to  the 


203-205. 


THE  POINT  IN  SPACE 


337 


axes  ;  then  projection  oi  d  =  P^P^  upon  these  lines  in  turn 
gives 

P^F^GOsa=P^X',  PlP2Cos^=Pl^^  F^P^cosy=P^Z\ 
^.e.,     c?cos  a  =  2-2  — a?i,         dcos  ^  =  ^2~  1/v       c?  cos  7=^2  — ^i? 
and,  finally, 


COSa= 


OCi}  —  OC-t 


cosp 


y^-Vi 


d     '    —-     d     '    «««Y  =  — ^—  •   •   •   L^J 

Tlifise    equations   give  the   required   direction   angles    of 

20s   The  point  which  divides  in  a  given  ratio  the  straight 

line  from  one  point  to  another.    Let 

z 

be  two  given  points,  and  let 
Pg  =  (a?3,  ^/g,  ^3)  be  a  third  point 
which  divides  the  line  P1P2  in  the 

given  ratio  — i,  so  that  ^^  =  — 1. 
m^  P^P^      ^^ 


m; 


2     F/ 


Fig.  145 


M, 


Let  P1P3  =  c?i,  and  P3P2  =  d^  ; 
then  by  Art.  204,  if  «,  /3,  7  be  the  direction  angles  of  P^P^-, 

•^1  **/0  ♦t/O  *t/Q  **^1 


COS  Ct  =  -^ 


d-i 


3. 


l_  ^ 


c?. 


^2        *^3 


C?r 


and 

Similarly, 
and  > 


i»3  = 


wiia?2  +  iW2a?i 


yo  _ 9 

m^z.2  +  in.yZi 

Zo  = — . 

nil  +  ^^2 


[8] 


It  will  be  noticed,   as  in  the  similar  problem  in  Part  I, 

Art.  30,  that  if  P3  divides  the  line  externally,  the  ratio  — 1 

???2 
must  be  negative  ;  and  the  above  formulas  still  apply. 

TAN.  AN.  GEOM.  — 22 


338 


ANALYTIC  GEOMETRY 


[Ch.  I. 


If  Pg  bisects  the  line  P^P^^  formulas  [8]  take  the  simpler 
forms 

^3- 2 '     ^^~ 2" — '     ^^ — 2 — *   •    •    •    L^J 


But 


206.  Angle  between  two  radii  vector es.  Angle  between  two 
lines.  Let  P^  =  (p^,  «^,  ^^,  7^)  and  P^  =  (p^,  a^,  P^,  73)  ^® 
two  given  points,  and  6  the  angle  included  by  the  radii  vec- 

tores  p-^  and  p^.  Then  the  pro- 
jections upon  OP^  of  the  line 
OP^  and  of  the  broken  line 
OMJSI^P^  are  equal  (Art.  17); 
hence, 

proj.   OP2  =  proj.  OM^M^P^, 
i.e.,     P2  cos  0  =0M2  cos'^i 
+  M^M^'  cos  ySi  +  M^'P^  cos  7i. 

OM2  =  P2  cos   «2i 

JfgMg'  =  P2  COS  72,  and  ifg' A  =  /'2  ^^s  72  ; 
hence, 

P2  COS  ^=/32  COS   ttg  cos  «i  +  /02  COS  ySg  COS  ^i-\-p2  COS  73  COS  7j  *, 
ie.,  cos  8  =  cos  ai  COS  a2  +  COS  Pi  COS  P2  +  COS  71  COS  72>  [1^] 

and  this  relation  determines  the  required  angle  0. 

It  follows,  since  any  two  straight  lines  in  space  have  their 
directions  given  by  the  direction  angles  of  radii  vectores 
which  are  parallel  to  them,  respectively,  that  formula  [10] 
applies  as  well  to  the  angle  6  between  any  two  straight  lines 
in  space,  whose  direction  angles  are  given. 

Two  special  cases  arise,  of  parallel  and  of  perpendicular 
lines.     If  the  two  given  lines  are  parallel,  evidently 

^^1  =  ^2'  Pi  =  P2'  vi  =  72;  [11] 


205-207.] 


THE  POINT  IN   SPACE 


389 


and  formula  [10]  reduces  to  eq.  [4].     If  the  lines  are  per- 
pendicular, cos  ^  =  0,  and  eq.  [10]  reduces  to 

cos  aj  COS  a2  -f  COS  Pi  COS  Pg  +  COS  Vi  COS  72  =  0.  .  .  .   [12] 


207.   Transformation  of  coordinates;   rectangular  systems. 

The  relations  found  in  Art.  202  to  exist  between  rectangu- 
lar and  polar  coordinates  of  a  point  may  be  used  as  formulas 
of  transformation  from  one  system  to  the  other  if  the  origin, 
the  pole,  and  the  reference  axes  are  coincident.  Two  other 
simple  transformations  may  be  useful,  (1)  from  one  set  of 
rectangular  coordinates  to  a  parallel  set,  i.e.,  a  change  of 
origin  only ;  and  (2)  from  one  set  of  rectangular  axes  to 
another  set  through  the  same  origin,  i.e.,  a  change  of  direc- 
tion of  axes.  Then  any  transformation  between  rectangular 
and  polar  systems  can  be  per- 
formed by  a  combination  of 
these  three  elementary  trans- 
formations. 

(1)  Change  of  origin  only. 
Let  the  new  origin  be  the  point 
0'  =  (^h,  k,  y);  then,  construct- 
ing the  coordinates  of  any 
point    P    with    reference     to 

each  set  of  coordinate  planes,  it  is  evident,  by  analogy  with 
Art.  71,  that 


Fig.  147 


oi>  =  a>'  +  h,    y-y'  +  h,    z  =  z'+J, 


[13] 


(2)   Change  of  direction  of  axes.     Let  a  second  set  of  rec- 
tangular axes,  OX',  0Y\  OZ',  have  the  direction  angles  «,, 

^v  Tr  ^T  Pv  yv  ''^^^^^  H-'  ^B^  73'  respectively,  with  the  old 
axes  OX,  OY,  OZ, 


340 


AJ^ALTTIC  GEOMETRY 


[Ch.  I. 


[14] 


Then  if  the  coordinates 
of  any  point  P  in  the  two 
systems  are 

X  =  OM, 
y  =  MM\ 
z  =  31' P, 
and         x'  =  OQ, 

y  =  QQ'. 

z'  =  Q'P. 

then  projections  of  OP  and  the  broken  line  OQQ'P  upon  OX, 
OY^  OZ^  in  turn,  will  be  equal;   hence, 

00  =  oc'  COS  aj  +  y'  COS  a^  +  z'  COS  a3, 
1/  =  a?'  COS  Pi  +  2/'  COS  po  +  z'  cos  P35 
z  =  x'  cos  7i  +  y'  cos  72  +  ^'  COS  73. 

These  formulas  are  for  transformation  from  the  first  sys- 
tem   to    the    second.     But,    also,    by    projecting    OP    and 
OMM'P  upon  0X\  0Y\   0Z\  respectively, 
x'  =  X  COS  «i  +  ?/  COS  (B^-\-  z  cos  7j, " 

y'  =z  X  cos  «i  +  ^  cos  13^  +  Z  COS  721  '  •  •  •   [15] 
z'  =  X  COS  «3  +  y  COS  ^83  +  2  COS  73, 

and  these  formulas  are  for  the  reverse  transformation,  from 
the  second  system  to  the  first. 

Note.  It  is  to  be  remembered  that  in  the  transformation  of  [14]  and 
[15],  twelve  conditions  exist,  by  eq.  [4]  and  eq.  [12],  three  of  each  of 
the  following  types, 

cos^ttj  +  cos^ag  +  cos'^ag  =  2, 

cos^a^  +  cos2/?j+  cos'^y^  =  1, 

cos  ttj^  cos  a.2  +  cos  (i^  cos  ^^  +  ^^^  Yi  ^os  yg  =  0, 

cos  a^  cos  ^j  +  cos  a^  cos  ^^  +  ^^s  a.^  cos  /Jg  =  0. 

These  equations  are  not  independent,  however,  but  reduce  to  six 
independent  equations. 


207].  THE  POINT  IN  SPACE  341 

It  is  clear,  by  reasoning  similar  to  that  of  Art.  75,  Part  I, 
that  none  of  the  transformations  [13],  [14],  and  [15],  neither 
separately  nor  in  combination,  can  alter  the  degree  of  an 
equation  to  which  they  may  be  applied. 

EXAMPLES    ON    CHAPTER    I 

1.  Prove  that  the  triangle  formed  by  joining  the  points  (1,  2,  3), 
(2,  3,  1),  and  (3,  1,  2),  in  pairs,  is  equilateral. 

2.  The  direction  cosines  of  a  straight  line  are  proportional  to  1,  2,  3 ; 
find  their  values. 

3.  Find  the  angle  between  two  straight  lines  whose  direction  cosines 
are  proportional  to  2,  2,  2,  and  5,  ~4,  7,  respectively. 

4.  The  rectangular  coordinates  of  a  point  are  (V3,  1,  2V3);  find 
its  polar  coordinates. 

5.  The  polar  coordinates  of  a  point  are  (8,  -,  ^j;   find  its  rectan- 
gular coordinates. 

6.  Express  the  distance  between  two  points  in  terms  of  their  polar 
coordinates. 

7.  Find    the    coordinates    of    the    points    dividmg   the   line  from 
(~2,  -3,  1)  to  (3,  ~2,  4)  externall}^  and  internally  in  the  ratio  2 : 5. 

8.  What  is  the  length  of  a  line  whose  projections  on  the  coordinate 
axes  are  4,  1,  3,  respectively? 

9.  Find  the  radius  vector,  and  its  direction  cosines,  for  each  of  the 
points  (-7,  1,  5),  (1,  -1,  -2),  (a,  0,  h). 

lOo   Find  the  center  of  gravity*  of  the  triangle  of  Ex.  1. 

11.  Find  the  direction  angles  of  a  straight  line  w^hich  makes  equal 
angles  with  the  three  coordinate  axes. 

12.  A  straight  line  makes  the  angle  30°  with  the  rc-axis,  and  75° 
with  the  2-axis.     At  what  angle  does  it  meet  the  ?/-axis  ? 

13.  Prove  analytically  that  the  straight  lines  joining  the  mid-points 
of  the  opposite  edges  of  a  tetrahedron  pass  through  a  common  point, 
and  are  bisected  by  it. 

14.  Prove  analytically  that  the  straight  lines  joining  the  mid-points 
of  the  opposite  sides  of  any  quadrilateral  pass  through  a  common  point, 
and  are  bisected  by  it. 

*  See  Ex.  15,  p.  42. 


CHAPTER   II 

THE  LOCUS  OF  AN  EQUATION.     SURFACES 

208.  Attention  lias  been  called  to  tlie  close  analogy 
between  the  corresponding  analytical  results  for  the  geom- 
etry of  the  plane  and  of  space.  It  is  evident  that  in 
geometry  of  one  dimension,  restricted  to  a  line,  the  point  is 
the  elementary  conception.  Position  is  given  by  one  vari- 
able, referring  to  a  fixed  point  in  that  line  ;  and  any  alge- 
braic equation  in  that  variable  represents  one  or  more  points. 
In  geometry  of  two  dimensions,  however,  it  has  been  shown 
that  the  line  may  be  taken  as  the  fundamental  element. 
Position  is  given  by  two  variables,  referring  to  two  fixed 
lines  *  in  the  plane  ;  and  any  algebraic  equation  in  the  two 
variables  represents  a  curve,  z.e.,  a  line  whose  generating 
point  moves  so  as  to  satisfy  some  condition  or  law.  Corre- 
spondingly, in  geoDietry  of  three  dimensions  the  surface  is  the 
elementary  conception.  Position  is  given  by  three  variables, 
referring  to  three  fixed  surfaces,  since  any  point  is  the  inter- 
section of  three  surfaces  ;  f  and  it  can  be  shown  that  any 
algebraic  equation  in  three  variables  represents  some  surface. 

*  Witli  polar  coordinates,  these  lines  are  a  circle  about  the  pole  with 
radius  =  p,  and  a  straight  line  through  the  pole  making  the  angle  6  with  the 
initial  line  (Art.  23). 

t  With  polar  coordinates,  these  surfaces  are  a  sphere,  about  the  origin  as 
center,  determined  by  the  radius  vector  p,  a  right  cone  about  the  ^-axis,  with 
vertex  at  the  origin,  determined  by  the  angle  ^,  and  a  plane  through  the 
s-axis  determined  by  the  angle  d  (Art.  201). 

342 


208-209.]  SUBFACES  343 

The  study  of  the  special  equations  of  first  and  second 
degree  will  be  taken  up  in  the  two  succeeding  chapters. 
Here  it  is  desired  to  show  that  an  algebraic  equation  in  three 
variables  represents  a  surface,  and  to  consider  briefly  two 
simple  classes  of  surfaces  :  (1)  cylinders,  ^.6.,  surfaces  which 
are  generated  by  a  straight  line  moving  parallel  to  a  fixed 
straight  line,  and  always  intersecting  a  fixed  curve  ;  and  (2) 
surfaces  of  revolution,  z.e.,  surfaces  generated  by  revolving 
some  plane  curve  about  a  fixed  straight  line  lying  in  its  plane. 

209.  Equations  in  one  variable.  Planes  parallel  to  coordi- 
nate planes.  From  the  definition  of  rectangular  coordinates, 
it  follows  that  the  equations 

a;  =  0,    ^  =  0,    2  =  0, 

represent  the  coordinate  planes,  respectively,  and  that  any 
algebraic  equation  in  one  variable  and  of  the  first  degree 
represents  a  plane  parallel  to  one  of  them.  Similarly,  an 
equation  in  one  variable  and  of  degree  n  will  represent  n 
such  parallel  planes,  either  real  or  imaginary.  For,  any  such 
equation,  as 

can  be  factored  into  7i  linear  factors,  real  or  imaginary, 

FoC^-^i)(^-^2)(--0(^-^J=  0;    ...    (2) 

and  by  the  reasoning  of  Part  I,  Art.  40,  eq.  (2)  will  repre- 
sent the  loci  of  the  n  equations 

X  —  x-^  =  0,    X  —  X2  =  0^    ••.,    X  —  x„  —  0, 

each  of  which  is  a  plane,  parallel  to  the  ^^-plane,  and  real  if 
the  corresponding  root  is  real.     In  the  same  way,  an  equa- 


344 


ANALYTIC  GEOMETRY 


[Ch.  II. 


tion  in  y  or  2  only  will  represent  planes  parallel  to  the  zx-  or 

Any  algebraic  equation  in  one  variable  represents  one  or 
more  planes  parallel  to  a  coordinate  plane. 

It  follows  at  once  by  Art.  39,  that  two  simultaneous 
equations  of  the  first  degree  in  one  variable  represent  the 
intersection  of  two  planes  parallel  to  coordinate  planes ; 
therefore,  represent  a  straight  line  parallel  to  the  coordi- 
nate axis  of  the  third  variable  ;  e.g..,  y  =  1.,  z  =  c,  considered 
as  simultaneous  equations,  represent  a  straight  line  parallel 
to  the  iT-axis. 

210.  Equations  in  two  variables.  Cylinders  perpendicular 
to  coordinate  planes.     Consider  the  equation 


2a; +  3^  =6, 


(1) 


with  two  variables  only.     In  the  a:?/-plane  it  represents  a 
straight  line  AB.      If,  now,  from  any  point  F  of  AB  a 


Fig. 149. 


straight  line  be  drawn  parallel  to  the  2-axis,  the  x  and  y 
coordinates  of  every  point  Q  on  this  line  will  be  the  same  as 
for  P,  and  therefore  satisfy  equation  (1).  Moreover,  if  the 
line  FQ  moved  along  AB^  and  always  parallel  to  the  2-axis, 


209-210.] 


SURFACES 


345 


still  the  coordinates  of  every  point  in  it  satisfy  equation  (1) . 
As  the  line  P^  is  thus  moved,  it  traces  a  plane  surface  per- 
pendicular to  the  o^^-plane  ;  and,  as  evidently  the  coordinates 
of  a  point  not  on  this  surface  do  not  satisfy  equation  (1) 
this  cylindrical  plane  is  the  locus  of  equation  (1). 
Again:  the  equation 

f^z'^=r^        .          .         .  (2) 

represents  in  the  ^^-plane  a  circle.  It  is  therefore  satisfied 
by  the  coordinates  of  any  point  $,  in  a  line  parallel  to  the 
^-axis,  through  any  point  P  of  this  circle ;  and  also  by 
the  coordinates  of  Q  as  this  line  PQ  is  moved,  parallel  to 


Fig.    150, 


the  X-axis  and  along  the  circle.  The  circular  cylinder  thus 
traced  by  the  line  PQ,  perpendicular  to  the  ^^-plane,  is 
the  locus  of  the  given  equation. 

Similarly,  it  may  be  shown  that  the  locus  of  the  equation 


^_  ^_  -J 


52 


(3) 


is  a  cylindrical  surface  traced  by  a  straight  line  parallel  to 
the  y-axis,  and  moving  along  the  hyperbola  whose  equation 
in  the  a;2-plane  is  equation  (3).  And,  in  general,  it  is  clear 
by  analogy  that  an^  algebraic  equation  in  two  variables  repre- 
sents a  cylindrical  surface  whose  elements  are  parallel  to  the 


346  ANALYTIC  GEOMETRY  [Ch.  n. 

axis  of  the  third  variable^  and  having  its  form  and  posi- 
tion determmed  hy  the  plane  curve  represented  by  the  same 
equation. 

As  a  direct  consequence,  it  is  clear  that  if  a  cylinder  lias 
its  axis  parallel  to  a  coordinate  axis,  a  section  made  by  a 
plane,  perpendicular  to  that  axis,  is  a  curve  parallel  to  and 
equal  to  the  directing  curve  on  the  coordinate  plane,  and  is 
represented  in  the  cutting  plane  by  the  same  equation. 
Thus,  the  section  of  the  elliptical  cylinder  whose  equation  is 
3  x^  -\-  y'^  =  5,  cut  by  the  plane  ;3  =  7,  is  an  ellipse  equal  and 
parallel  to  the  ellipse  whose  equation  is  3  a;^  +  ^^  =  5. 

211.  Equations  in  three  variables.  Surfaces.  A  solid 
figure  has  the  distinctive  property  that  it  can  be  cut  by  a 
straight  line  in  an  infinite  number  of  points,  while  a  sur- 
face or  line  can,  in  general,  be  cut  in  only  a  finite  number. 
A  line  has  the  distinctive  property  that  it  can  be,  in  gen- 
eral, cut  by  a  plane  in  only  one  point,  while  a  surface  may 
be  cut  in  a  curve.  To  show  that  the  locus  of  an  algebraic 
equation  in  three  variables  is,  in  general,  a  surface,  it  is  suf- 
ficient to  show  that,  in  general,  a  plane  will  cut  it  in  a  curve, 
while  a  straight  line  will  cut  it  in  a  finite  number  of  points. 

Let  the  given  equation  be 

f(x,y,z)=^,       .         .         .         (1) 

and  let  z  =  c         .         ,         .         (2) 

be  a  plane  parallel  to  the  a^^-plane.     The  points  of  inter 
section  of  these  two  loci  will  be  on  the  locus  of  the  equation 

f(x,  y,  c)=0;       .  .  .  (3) 

and,  by  Art.  210,  they  lie,  therefore,  upon  a  plane  curve,  cut 
from  the  cylinder  whose  equation  is  (3),  by  the  plane  whose 
equation  is  (2).  Hence  the  locus  of  equation  (1)  is  not  a  line. 


210-212.]  -       SURFACES  347 

Again,  let  ^  z=  h,     z  =  o        .         .         .         (4) 

be  the  equations  of  a  straight  line  (Art.  209),  parallel  to  the 
a;-axis.  The  points  of  intersection  of  locus  (1)  and  the  line 
(4)  will  be  also  on  the  locus  of  the  equation 

'Z=.^  X^,  J,  0=0;      .        .        .        (5) 

which,  since  the  equation  is  in  one  variable,  of  finite  degree, 
will  represent  a  finite  number  of  planes  parallel  to  the  ^z- 
plane,  and  therefore  having  a  finite  number  of  points  of 
intersection  with  the  line  (4).  Hence  the  locus  of  equation 
(1)  is  not  a  solid. 

Therefore,  the  locus  of  any  algebraic  equation  in  three  vari- 
ables is  a  surface. 

212.  Curves.  Traces  of  surfaces.  Two  surfaces  intersect 
in  a  curve  in  space  ;  and  since  every  algebraic  equation  in 
solid  analytic  geometry  represents  a  surface,  a  curve  may  be 
represented  analytically  by  the  two  equations,  regarded  as 
simultaneous,  of  surfaces  which  pass  through  it.  Thus  it 
has  been  seen  that  the  equations  y  =  b^  z  =  c  separately  rep- 
resent planes,  but  considered  as  simultaneous  represent  the 
straight  line  which  is  the  intersection  of  those  planes.  But 
by  the  reasoning  of  Art.  41,  the  given  equations  of  a  curve 
may  be  replaced  by  simpler  ones  which  represent  other  sur- 
faces passing  through  the  same  curve.  In  dealing  with 
curves  it  is  often  useful  to  obtain,  from  the  equations  given, 
equations  of  cylinders  through  the  same  curve ;  z.e.,  it  is 
generally  useful  to  represent  a  curve  by  two  equations  each 
in  two  variables  only. 

Example  :  The  curve  of  intersection  of  the  two  surfaces, 

(1)  a;2  +  ?/2  ^  ^2  _  25  =  0     and     (2)  x"^  +  3/2  -  16  =  0, 


348  ANALYTIC  GEOMETRY  [Ch.  II. 

is  also  the  intersection  of  the  surfaces 

a;2  +  ^2  +  ;22  _  25  -  (x2  +  2/2  -  16)  =  0,  i.e.,  z=±^,  (3) 

with  the  surface  (2).     The  curve  is  therefore  composed  of  two  circles  of 
radius  4,  parallel  to  the  a:y-plane  at  distances  +  3  and  —  3  from  it. 

Conversely,  the  curves  of  intersection  of  a  surface  with 
the  coordinate  planes  may  be  used  to  help  determine  the 
nature  of  a  surface.  These  curves  are  called  the  traces  of 
the  surface. 

Thus,  the  surface  x^  +  ^^  +  ^^  _  25  has  the  traces 

on  the  ^^-plane,  where  a;  =  0,  ?/2  -|-  ^^  =  25 ; 
On  the  2a;-plane,  where  y  =  0,  a:^  +  ^2  _  25 ; 
on  the  a:^-plane,  wdiere  ^  =  0,  x^-\-  y'^=  25. 

Each  of  these  traces  is  a  circle  of  radius  5,  about  the 
origin  as  center ;  the  surface  is  a  sphere  of  radius  5  with 
center  at  the  origin. 

Since  three  surfaces  in  general  have  only  one  or  more 
separate  points  in  common,  the  locus  of  three  equations,  con- 
sidered as  simultaneous,  is  one  or  more  distinct  points. 

213.  Surfaces  of  revolution.  Analogous  to  the  cylinders 
are  the  surfaces  traced  by  revolving  any  plane  curve  about 
a  straight  line  in  the  plane  as  axis.  From  the  method  of 
formation,  it  follows  that  each  plane  section  perpendicular 
to  the  axis  is  a  circle,  —  the  path  traced  by  a  point  of  the 
generating  curve  as  it  revolves ;  and  the  radius  of  the  circle 
is  the  distance  of  the  point  from  the  axis  in  the  plane  before 
revolution  begins.  These  facts  lead  readily  to  the  equation 
of  any  surface  of  revolution,  as  a  few  examples  will  show. 

(a)    The  cone  formed  hy  revolving  about  the  z-axis  the  line 

2a;+ 3^  =  15.        .         .         .         (1) 


212-213.] 


SURFACES 


349 


Any  point  P  of  the  line  (1)  traces  during  the  revolution 
a  circle  of  radius  ZP,  parallel  to  the  a:?/-plane.  The  equa- 
tion of  that  path  is 

x^  +  y'-^  ZPI 
z 


Fig.  151. 


But  in  the  a;s-plane,  before  reA^olution  is  begun,  LP  is  the 
abscissa  x  oi  P  ;  hence,  by  equation  (1), 

15-32! 


LP  =  x  = 


2 


so  that  the  equation  of  the  path  of  P  is 


a?  +  f=(l^jzlll\ 


(2) 


But  P  is  any  point  of  line  (1);  hence  equation  (2)  is  sat- 
isfied by  every  point  of  the  line,  and  represents  the  surface 
generated  by  the  line,  which  is  the  required  conical  surface. 
(^)    The    sphere  formed  by  revolving  about  the  y-axis  the 

circle 

2:2-1-^2^25.  .  .  .  (3) 

In  this  case,  any  point  P  of  the  curve  traces  during  the  revo- 
lution a  circle  of  radius  iVP,  parallel  to  the  2a:-plane.  The 
equation  of  this  path  is  therefore 


a;^ 


-j-  ^2  ^  JSfP^^ 


350 


ANALYTIC  GEOMETBT 


[Ch.  n. 


Fig.  152. 


But    in    the    ^ry-plane,    by 
equation  (3) 

NP  =  x=  V25^2. 

Hence,  substituting  above, 

2^2  _|_  ^2  _   25  _  ^2^ 

i.e.,     x^+if  +  z^=2b',     (4) 

wliich  is  the  equation  of  the 
required  spherical  surface. 

((?)  The  surface  formed 
hy  revolving  about  the  x-axis 
the  curve 


2=(a;-l)(a:-2)(2^-3)  [cf.  Art.  37].    ...    (5) 

Any  point  P  of  the  generating  curve  traces  a  circle  parallel 
to  the  ?/2 -plane,  with 
a  radius  MP  equal  to 
the  2-abscissa  in  equa- 
tion (5).  Hence  the 
equation  of  its  path  is 


o 


B  cW  \M 


-X 


y2  -I-  ^2  ^  ]^^ 

i.e.,  if  -\-  z^  =  (x  ~  1)2 
(2:-2)2(a^-3)2;  .  .  .(6) 

which  is  the  equation 
of  the  required  surface. 
(67)  Of  the  various 
surfaces  of  revolution 
those  of  particular  interest  are  generated  by  revolving 
about  their  axes  the  various  conic  sections,  giving  the 
cones,  spheres,  paraboloids,  ellipsoids,  and  hyperboloids  of 
revolution. 


/ 


y 


Fig.  153. 


213.]  SUBFACES  351 

The  student  may  verify  the  equations  of  the  following 
surfaces  ;  * 

The  sphere  :  with  center  at  the  point  (a,  5,  (?),  and  radius  r, 

(^^-ay-\-(i/-by-i-(z-cY  =  r^;     ...     (7) 

with  center  at  the  origin,  and  radius  r, 

^2  +  ?/2  4-  2;2  =  r\         ...        (8) 

The  cone:  the  surface  generated  by  the  right  line  z  =  mx-\-c, 
rotated  about  the  ^-axis, 

x^-\-f  =  ^^  -/)'.        ....     (9) 

The  oblate  spheroid :    the  surface  generated  by  the  ellipse 

—  +  —  =  1,  rotated  about  the  minor  axis, 
a^     b^ 

/Y>u  01^  ^^ 

-2  +  ^-2  +  h=^-     ■     •     •     (10) 

a^      a'^      0'^ 
The  prolate  spheroid :  the  surface  generated  by  the  ellipse 

—  +  —  =  1,  rotated  about  the  major  axis, 

/v>4  /}/^  n2t 

'       1  +  1  +  ^=1-  •         •         •  (") 

The  hyperboloid  of  one  nappe :  the  surface  generated  by 

the  hyperbola  —  —  ^  =  1,  rotated  about  the  conjugate  axis, 
a^      0^ 

/yii  nia  /ya 

^  +  ^-1=1-        •       •       •        (12) 
a^      a^      0^ 

The  hyperboloid  of  two  nappes :  the  surface  generated  by 

the  M^perbola  7-,  —  '-^  =  1?  rotated  about  the  transverse  axis, 
0^      a^ 

/yta         01^         ^y^ 

*  See  Chap.  IV,  where  diagrams  are  given  for  the  corresponding  cases 
of  the  general  quadric,  with  elliptical  instead  of  circular  sections. 


352  ANALYTIC   GEOMETRY  [Ch.  II.  213. 

The  paraboloid  of  revolution :  the  surface  generated  by  the 
parabola  x^  =  -^  pz^  rotated  about  its  axis, 

x^  -\-  y^  =  ^  pz.         .         .        .        (14) 

EXAMPLES    ON    CHAPTER    II 

What  is  the  locus  of  each  of  the  following  equations? 

1.  ^2  -  6  a:  +  9  =  0.  4.    aj;2  +  hxy  +  cy^  =  0. 

2.  2a:  +  4  =  0.  5.   4:yz  +  6  ij  -  Sz  +  1  =  0. 

3.  x^-2xy  +  y'^-\-2x-2y  +  l=0.      6.    z'^-9y  =  9. 

What  are  the  curves  of  intersection  of  the  surfaces  represented  by 
the  equations 

7.  ?/  +  3  =  0,     3x2 +  3^2 +  3^22  ^20? 

8.  x^  -y^  =  0,     z  =  a'> 

9.  x2  +  ?/2  +  2-2  ^  9,     4:r2  +  ?/2  =  4? 

10.  9(z2  +  y2)_  .2^o5_10~,     5  =±5? 

11.  3:r2-4y2_~2^i2,     ^  +  |^!=1? 

Determine  the  projections  upon  the  coordinate  planes  of  the  following 
surfaces : 

12.  a;2  +  ?/2  +  4  s2  =  25  ;  13.   Sx^  -  4:y^  -  z^  =  12. 

Find  the  equation  of 

14.  the  paraboloid  of  revolution  one  of  whose  traces  is  ?/2  =  —  5  x  +  3. 

15.  the  curve  of  revolution  one  of  whose  traces  is  y  =  —  5  x  +  3  and 
whose  axis  is  the  axis  of  y.     Find  its  vertex. 

16.  the  oblate  spheroid  one  of  whose  traces  is  —  +  —  =  1. 

<-       o 

V        z 

17.  the  prolate  spheroid  one  of  whose  traces  is  ^  +  — -  =  1. 

18.  the  surface  of  revohition  whose  axis  is  the  axis  of  x  and  one  of 
whose  traces  is  x'^y  —  1  =  0. 

19.  the   hyperboloid  of   two   nappes   one   of  whose   traces   is   16  x^ 

-9^2:=!. 

20.  the    sphere    described    about    the    major    axis    of    the    ellipse 
4  a:2  +  9  3/2  —  24  a;  =  0  as  diameter. 


CHAPTER   III 

EQUATIONS   OF   THE   FIRST   DEGIJEE 
Aoc  +  Bij  +  Cz  +  D  =  () 

PLANES  AND   STRAIGHT  LINES 

I.    The  Plane 

214.  Every  equation  of  the  first  degree  represents  a  plane. 
A  plane  is  a  surface  such  that  it  contains  every  point  on  a 
straight  line  joining  any  two  of  its  points. 

Let  P^  =  (a;^,  ?/-^,  2j)  and  P^  =  (x^,  y^^  z^  be  any  two  points 
of  the  surface  whose  equation  is 

^ic  + J5t/  +  Cs  +  1>  =  0,      .       .       .     [16] 

so  that  Ax^  -\-  By^  +  Cfe^  +  i>  =  0      .      .      .       (1) 

and  Ax^  +  By^  +  0b^  +  D  =  0.     .      .      .      (2) 

Now,  if  Pg  =  (2^3,  ^3,  ^3)  be  any  point  on  the  straight  line 
from  Pj  to  P^  at  a  distance  d^  from  P^  and  d^  from  P^^  then, 
by  Art.  205, 

^  "~        C^i  +  6?2      ^  C?l  +  t?2      '        ^  ~       t^l  +  d^  ^'  ^ 

But  this  point  lies  on  the  surface  represented  by  equation 
[16];  for,  substituting  its  coordinates  from  (3)  in  equation 
[16],  the  latter  becomes 

Cti  -p  Clt)  Cl-i  -f"  (to 

TAX.   AN.   GEOM. 23  353 


354  ANALYTIC  GEOMETRY  [Ch.  III. 

which  is  a  true  equation,  since  each  parenthesis  vanishes 
separately  by  equations  (1)  and  (2).  Hence  every  point  of 
the  line  P^P'^  is  on  the  locus  of  equation  [16],  and  that 
locus  is  therefore  a  plane.  Every  algebraic  equation  of  the 
first  degree  in  three  variables  represents  a  plane. 

215.  Equation  of  a  plane  through  three  given  points.     The 

general  equation  of  the  first  degree, 

Ax-vBy-{-Cz  +  D  =  Q,       .      .      .       (1) 

has  only  three  arbitrary  constants,  viz.  the  ratios  of  the 
coefficients.     If  three  given  points  in  the  plane  are 

then  these  ratios  may  be  found  from  the  three  equations, 

Ax^  +  By^  +  Cfej  +  i>  =  0,  ■ 

Ax^  +  %2  +  ^^2  +  -^  =  0,  •      .      .      .      (2) 

Ax^  +  %3  +  C^3  +  2)  =  0, . 

considered  as  simultaneous. 

In  solving  equation  (2)  for  the  required  ratios,  two  special 

cases  may  occur  :    {a)  The  value  of  one  of  the  coefficients 

may  be  zero,  then  the  ratios  determined  must  not  have  that 

coefficient   in   the    denominator.  E.g..,    if    J>  =  0,    solution 

A      Ti  n  A     Ti 

should   not   be  made  for  — ,    —,  — ,  but  for  — ,  — •  (say). 

jj     U  JD  0      0 

(5)  The  equations  may  differ  only  by  constant  factors,  then 
the  three  equations  have  an  infinite  number  of  solutions. 
This  is  explained  by  the  fact  that  the  points  are  on  a  straight 
line,  and  any  plane  through  the  line  will  pass  also  through 
the  points. 

216.  The  intercept  equation  of  a  plane.     A  plane  will  in 
general  cut  each  coordinate  axis  at  some  definite  distance 


214-217.] 


PLANES  AND   STRAIGHT  LINES 


355 


from  the  origin,  and  this  distance  is  called  the  intercept  of 
the  plane  on  the  axis.  If  a,  6,  c  be  the  intercepts  on  the  x-, 
?/-,  and  2-axes,  respectively,  of  the  plane  whose  equation  is 

Ax  +  By  +  (7^  +  i>  =  0,      .       .       .       (1) 

then  the  points  (a,  0,  0),  (0,  ^,0),  (0,  0,  c)  are  points  of  the 
plane,  and  therefore  (cf .  Art.  215) 

J.a  +  i>  =  0,     Bh  +  D  =  0,     Cc^D  =  0, 

B 

•  •         • 

G 

Hence  equation  (1)  may  be  written 


i.e. 


a 


0 


(2) 


I.e., 


a         h         c 


-  +  ^  +  --1- 


[17] 


and  this  is  the  equation  of  the  plane  in  terms  of  its  intercepts. 

217.  The  normal  equation  of  a  plane.  A  plane  is  wholly 
determined  in  position  if  the  length  and  direction  be  known 
of  a  perpendicular  to  it 
from  the  origin ;  and  this 
method  of  fixing  a  plane 
leads  to  one  of  the  most 
useful  forms  of  its  equa- 
tion. Let  OQ  be  the 
perpendicular  from  the 
origin  0  to  the  plane 
ABO^  let  p  be  its  length, 
always  considered  as 
positive,  and  let  «,  /3,  7  y 
be  its  direction  angles.  Let  P  =  (x,  ?/,  z)  be  any  point  of 
the  plane,  and  draw  its  coordinates  OM,  MM',  M'P.  Then, 
projecting  upon  OQ^ 


Fig.  154 


356  ANALYTIC  GEOMETRY  [Ch.  III. 

proj.  OMM'P  =  proj.  OP, 
hence  proj.  0M+  proj.  MM'  -\-  proj.  M' P  =  proj.  OP, 
that  is,  a?cosa  +  t/cosp  +  2:  cos-y  =i>.     .      .      .      [18] 

This  is  called  the  normal  equation  of  the  plane. 
There  are  two  special  cases  to  be  considered  : 

(1)  If  the  plane  is  perpendicular  to  a  coordinate  plane, 
e.g.,  to  the  :?^y-plane  (cf.  Art.  210),  then  7  =  90°,  cos  7=0, 
and  equation  [18]  reduces  to 

a;  cos  «  +  ?/ cos /3  =  j9.       .       .       .       [19] 

(2)  If  the  given  plane  is  parallel  to  one  of  the  coordinate 
planes,  e.g.,  to  the  a:y-plane  (cf.  Art.  209);  then  «=/3=90°, 
7  =  0°,  and  eq.  [17]  reduces  to 

z=p.  .  .  .  [20] 

218.  Reduction  of  the  general  equation  of  first  degree  to  a 
standard  form.*  Determination  of  the  constants  a,  b,  c,  i>, 
a,  p,  7.  I.  Intercept  form.  In  Art.  216  a  method  has  been 
indicated  for  reducing  the  general  equation 

Ax  +  By^Cz  +  D=0       .       .       .       (1) 

to  the  intercept  form.  Since  the  points  (a,  0,  0),  (0,  h,  0), 
and  (0,  0,  c)  are  on  the  plane  (1),  it  follows  that  the  inter- 
cepts are 

D.I)  D  .0. 

"  =  -T    ^  =  -B^    '  =  -c'   •    •    •   (^) 

II.    Normal  form.     If  equation  (1)  and  the  equation 

X  cos  a  -\-  y  cos  fi  -{•  z  cos  7  —  _p  =  0    .     .     .     (3) 
represent  the  same  plane,  then  their  first  members  can  differ 

*  The  reduction  of  this  article  gives  a  second  proof  that  the  general  alge- 
braic equation  of  first  degree  always  has  for  its  locus  a  plane. 


217-219.]  PLANES  AND   STRAIGHT  LINES  357 

only  by  a   constant  factor,  m  (cf .  Art.  203,  eqs.  [5] ;  also 
Art.  58); 

therefore 

mA  =  cos  a,     7nB  =  cos  yS,     m  0  =  cos  7,     mD  =  —  p, 

but,  by  [4],         cos^  «  +  cos^  jS  +  cos^  7  =  1, 
hence        m^(^A'^  -{-  B^  +  0^}  =  1,  and  m  = 


VWTW+c^ 


Then     cosa^- ^  cosp  = ^ 


V^2  +  ^2  +  c^2  V^2  +  ^2  +  (72 

r'  —  /> 

cos  7  =  >  p 


V^2  +  ^2  +  ^.2  V^2  _j.  j52  4.  (72 

Equation  (1)  written  in  the  normal  form  is  then 
A  ,  B 


[21] 


g=    ,  _  ;  ...    (5) 


VA2  +  ^2  +   C'2  VJ.2  +  ^  +    6'2 

therefore,  to  reduce  equation  (1)  to  the  normal  form,  it  is  nec- 
essary only  to  transpoBe  the  constant  term  to  the  second  mem- 
ber of  the  equation^  and  then  divide  both  members  by  the  square 
root  of  the  sum  of  the  squares  of  the  coefficients  of  the  variable 
terms.  The  sign  of  the  radical  is  determined  by  the  fact 
(Art.  217)  that  p  is  taken  positive  ;  hence,  the  sign  of  the 
radical  is  the  opposite  of  the  sign  of  the  constant  term. 

219.  Tlie  angle  between  two  planes.  Parallel  and  perpen- 
dicular planes.  The  angles  formed  by  two  intersecting 
planes  are  the  same  as  the  angles  formed  by  two  straight 
lines  perpendicular  to  them  respectively;  i.e.^  is  the  same 


358  ANALYTIC  GEOMJ^TRT  [Ch.  IIL 

as  the  angles  between  the  respective  normals  from  the  origin 
to  the  planes.     If 

A^x  +  B^y  +  O^z  +  I)^  =  0,     .      .      .      (1) 

and  A^x  +  B^y  +  C^z  +  i>2  =  0,     .      .      .      (2) 

be  two  planes,  then  the  direction  cosines  of  their  normals 
are  respectively  (eqs.  [21]) 

COS«i  =  —  ^^       ,  C0S)8i= —  ^       ,  C0S7l=-  ^ 


V^i2+^i^+CV  y/A{^  4-  Bi^  +  Ci'^  V^i^ + Bi^  +  C{^ ' 

cosctg^  '^    ,  etc., 

y/A^^+Bi^+Ci^ 

and  by  equation  [4],  if  0  be  the  angle  betAveen  the  two  planes, 
and  hence  between  the  two  normals, 

cos  9  33  ^l^2  +  Blg2jLCie2  .     .     .     p22-| 

There  are  two  cases  of  special  interest. 

I.  Parallel  planes.  If  the  planes  (1)  and  (2)  are  parallel, 
their  normals  from  the  origin  will  have  the  same  direction  co- 
sines, and  differ  only  in  length  ;  therefore,  by  equations  [20], 
the  equations  of  the  planes  must  be  such  that  the  coefficients 
of  the  variable  terms  are  the  same  in  the  two  equations,  or 
can  be  made  the  same  by  multiplying  one  equation  by  a 
constant.  In  other  words,  if  the  planes  (1)  and  (2)  are 
parallel,  then 

X  =  f  =  ^'        •        •        •        [23] 

-^2       ^2       ^2 

and  the  plane         Ax  -\-  By  -{-  Cz  -\-  K=  0      .       .       .       (3) 
is  parallel  to  the  plane 

Ax  +  By  +  Cz  +  B  =  0,  .      .       (4) 

for  all  values  of  the  parameter  K. 


219-221.]  PLAJ^ES  AND   STRAIGHT  LINES  359 

II.  Perpendicular  planes.  If  the  planes  (1)  and  (2)  are 
perpendicular  to  each  other,  then  cos  ^  =  0, 

and  AiA2  + BiB.^+ CiC2  =  0;     .       .       .       {24:'] 

and  conversely. 

220.  Distance  of  a  point  from  a  plane.     Let 

A  =  C'^i'^i'^i) 

be  a  given  point,  and 

Ax-{-B^-{-Cz-\-D=^0      .       .       .       (1) 

a  given  plane.  The  perpendicular  distance  of  P^  from  the 
plane  is  equal  to  the  distance  from  the  plane  (1)  to  a  parallel 
plane  through  the  point;  i.e.,  is  equal  to  the  difference  in 
the  lengths  of  the  normals,  from  the  origin,  to  these  two 
parallel  planes. 

The  parallel  plane  through  Pj  has  for  its  equation  by 
Art.  219,  equation  (3), 

Ax  +  Bt/  -\-0z  =  Ax^  +  By^  +  Cz^     .    .    .    (2) 

By  [21],  the  lengths  of  the  normals  to  planes  (1)  and  (2) 
are,  respectively, 

^^  -I>  ,  _  Ax,  +  By,  +  Cz^ 

therefore  ii  d  =  p'  —  p  hQ  the  required  distance, 

a  =  ^^^i±^?yi±j^^i±R.    .    .    .    [25] 

V^2  +  B^  +  C^ 

In  formula  [25],  the  sign  of  the  radical  is  taken  opposite 
to  the  sign  of  D  (Art.  218)  ;  and  the  sign  of  d  shows  on 
which  side  of  the  given  plane  lies  the  given  point. 

II.   The  Straight  Line 

221.  Two  equations  of  the  first  degree  represent  a  straight 
line.      Every  equation   of  first   degree   represents  a   plane 


360  ANALYTIC  GEOMETRY  [Ch.  III. 

,(Art.  214),  and  two  equations  considered  as  simultaneous 
represent  the  intersections  of  their  two  loci  (Art.  39). 
Therefore  since  tw^o  planes  intersect  in  a  straight  line,  the 
locus  of  the  two  simultaneous  equations  of  first  degree, 

A^x  +  B^y  +  C^z  +  D^  =  Q,  A^x  + B^y +  C^z  +  D^  =  0, .  .  .  (1) 

is  a  straight  line.  As  suggested  in  Art.  212,  it  is  generally 
more  simple  to  represent  the  straight  line  by  equations  in 
two  variables  only,  standard  forms^  to  which  equation  (1) 
can  always  be  reduced. 

222.   Standard  forms  for  the  equations  of  a  straight  line. 

(a)  The  straight  line  through  a  given  point  in  a  given  direction. 
Let  P^  =  (x^^  ?/j,  z^  be  a  given  point,  and  a,  y8,  7  the  direc- 
tion angles  of  a  straight  line  through  it.  Let  P  =  (x^  y,  2) 
be  any  point  on  the  line,  at  a  distance  d  from  Py  Then  by 
equation  [6], 

d  cos  a  =  X  —  x-^^  d  cos  P  =  y  —  yi-,  d  cos  ry  —  z  —  z-^^  .  .  .  (1) 

hence  x^^oo,^y-j^^z-^        _       _       _       ^^^-^ 

COS  a  COSp  COS7 

which  are  the  equations  of  a  straight  line  in  the  first  standard 
form,  called  the  symmetrical  equations. 

(6)  The  straight  line  through  two  given  points.  Let  P^  = 
(xy  y^  z^  and  P^  =  {x^.,  y^,  z^  be  the  given  points.  Any 
straight  line  passing  through  P^  has  [26]  for  its  equations. 
If  the  line  passes  also  through  P^^  then 

^2  ~  ^1  _  Vi  ~  Vx  _  ^2  ~  ^1  .  fs\\ 

cos  a    ~    cos  y8    ~~    cos  7  '     *      *      *      ^  ^ 

and  hence  from  equations  [26]  and  (2),  by  division  to 
eliminate  the  unknown  direction  cosines. 


a?2  -  ^1     2/2  -  2/1     ^2  -  ^1 


[27] 


221-222.] 


PLANES  AND   STRAIGHT  LINES 


361 


These  are  the  second  standard  forms  for  the  equation  of  a 
straight  line. 

(c)  The  straight  line  ivith  given  traces  on  the  coordinate 
planes.  One  of  the  simplest  set  of  planes  for  determining  a 
straight  line  is  a  pair  of  planes  through  the  line  and  perpen- 
dicular respectively  to  the  coordinate  planes  (cf.  Art.  212). 
Then  the  equation  of  these  planes  will  be  the  same  as  the 
equations  of  the  traces  of  the  line  on  the  corresponding  coor- 
dinate planes  (Art.  210).  Thus,  if  the  equation  of  the  traces 
of  a  given  line  upon  the  zx-  and  ^2-planes  are,  respectively, 


y  =  nz  -\-  d^ 

then,  considered  as  simultaneous,  these  are  also  the  equa- 
tions of  the  given  line  in 
space. 

In  Fig.  155  the  given 
traces  are  ABL'  in  the 
0a;-plane,  and  CDN'  in  the 
2/2-plane  ;  P  is  any  point 
in  the  given  straight  line, 
and  §,  i2,  S  are  the  points 
where  the  line  pierces  the 
xy-^  yz-.,  2!a;-planes,  respec- 
tively. Then  it  is  clear 
that  in  equations  [28] 

m  =  tanZ  O^^ji.    h  =  OB, 
n  =  tanZ  OOB,    d=  OB. 


Fig.  155 


(3) 


Also,  since,  by  equations  [28], 

bn 


m  m 


00=-i,   OS  =  ^"  -  '^"', 

n  n 


362  ANALYTIC  GEOMETRY  [Ch.  III. 

therefore  the  points  where  the  given  line  pierces  the  coordi- 
nate planes  are 

Q^(_b,d,0),  B^fo,  *?i=*!!,_A\  fef*Ji:z*^,  0,  -^.  (4) 
^  \         m  mj  \      n  nj 


223.  Reduction  of  the  general  equations  of  a  straight  line 
to  a  standard  form.  Determination  of  the  direction  angles 
and  traces. 

I.  Third  standard  form:  traces.  The  traces  of  a  straight 
line  have  the  same  equations  as  have  the  planes  of  projec- 
tion of  the  straight  line  upon  the  coordinate  planes,  respec- 
tively. They  may  be  obtained,  therefore  (Art.  210),  by 
eliminating  in  turn  each  of  the  variables  ^,  ^,  x  from  the 
given  equations. 

This  may  be  illustrated  by  a  numerical  example. 

Given  the  equations 

3a;  + 2^  +  2 -5  =  0,  a:-f  2^ -2^  =  3,      .      .      .      (1) 

representing  a  straight  line.  Eliminating  z,  y,  and  x,  successively,  the 
equations 

7:r  + 4?/ -13  =  0,  2a; +  32 -2=0,  4^-7^-4  =  0  ...  (2) 

are  obtained,  each  representing  a  plane  through  the  given  line  and  per- 
pendicular to  a  coordinate  plane.  Therefore  these  equations  are  also  the 
equations  of  the  traces  of  the  line,  in  the  xy-,  zx-,  and  i/s-planes,  respectively. 

II.  First  standard  form :  direction  angles.  The  method  of 
reducing  the  general  equations  of  a  straight  line  to  the  first 
standard  form,  and  finding  its  direction  angles,  can  also  be 
illustrated  by  a  numerical  case. 

Considering  still  the  line  whose  equations  are  (1)  above,  and  w^hose 
traces  are  given  by  equations  (2) ;  and  taking  the  equations  of  any  two 
of  its  traces,  e.g.., 

2a:  +  3s-2  =  0,     42/-7s-4  =  0;      .       .       .       (3) 


222-224.]  PLANES  AND   8TEAIGHT  LINES  363 

these  have  one  variable,  z,  in  common.     Equating  the  values  of  this 
common  variable  from  the  two  equations,  gives 

_  -2.r  +  2_4y-4 
3  7 

which  may  be  written,  to  correspond  with  equations  [26], 

g  -Q  ^  a;-  1  ^  y  -  1  ^  ^  /^A 

1  3  7        *  •  •  •  V    y 

i  —   2  ? 

Now,  although  the  denominators  1,  —  f,  |  of  equation  [4]  are  not 
du'ection  cosines  of  any  line,  yet,  by  equations  [5],  they  differ  from 
such  direction  cosines  only  by  the  factor 

Vl  +  f  +  ft  =  iVlOl. 
Rewriting  equations  (4)  in  the  form 

X  —  1     y  —  ^     z  —  0 


6 


(5) 


loi     VlOl     VlOl 


it  corresponds   entirely  to  equations   [26].     Therefore  the   line   passes 
through  the  point  (1,  1,  0),  and  its  direction  angles  are  given  by  the 

relations 

6  o  7  ^ 

cos  a  = ,    cos  p  =  — ,    cos  y 


VlOl  VlOl  VlOl 

The  method  given  above  is  evidently  perfectly  general. 

224.  The  angle  between  two  lines ;  between  a  plane  and  a 
line.  If  the  equations  of  two  straight  lines  be  written  in  the 
form 

^  -  ^1  ^  ,y  -  ^1  ^  ^  -  ^1^      .       .       .       (1) 

^  —  ^1  ^y  —  Vi  ^^  —  H      ...      (2) 

«2  ^2  ^2 

then  by  Art.  223,  II,  their  direction  cosines  are,  respectivel}^ 
cos  «j  =  —  t  cos  ^2  = 


cos  /3i  =  ^  etc.,      ...     (2) 


364  ANALYTIC   GEOMETRY  [Ch.  111. 

and  therefore,  by  equation  [10],  the  angle  between  the  two 
lines  is  given  by  the  equation 

coGe=  ^1^.2  +  &1&2  +  C1C2  |-29] 

Again,  the  angle  between  the  straight  line 

^'E^zJb.  =  y-::zli  =  ^^-zJi\^     .      .      c     (3) 
a  h  c 

and  the  plane 

Ax^-By-\-  Cz^I)=^       .       .       .       (4) 

is  the  complement  of  the  angle  between  the  line  (3)  and  the 
perpendicular  to  the  plane  (4)  from  the  origin.  Therefore, 
by  equations  [10]  and  [21],  and  Art.  223,  II,  the  required 
angle  is  given  by  the  equation 

sine^ aA+bB  +  cC     ^ 

Conditions  for  perpendicularity  and  parallelism  precisely 
like  those  of  Art.  219  may  be  obtained  from  equations  [29] 
and  [30]. 

EXAMPLES    ON    CHAPTER    III 

1.  Find  the  equation  of  a  line  through  the  points  (1,  2,  3)  and 
(3,  2,  1). 

2.  Find  the  equation  of  a  plane  through  three  points  (1,  2,  3), 
(3,  2,  1),  and  (2,  3,  1). 

3.  Write  the  equations  of  the  straight  line  through  the  point 
(1,  2,  3),  and  having  its  direction  cosines  proportional  to  V-S,  1,  2V3. 

4.  What  are  the  traces  of  the  line  of  Ex.  1  upon  the  coordinate 
planes?    Where  does  the  line  pierce  those  planes? 

5.  Find  the  equations  of  a  straight  line  through  the  point  (1,  2,  3) 
and  perpendicular  to  the  plane  x-\-2y  +  ^z  =  6. 

Reduce  to  the  intercept  and  normal  forms,  and  determine  which 
octant  each  plane  cuts  : 

6.  2x-^y-z  =  7;  7.   5y  +  2z  -1  =  x.     - 


224.]  PLANES  AND   STRAIGHT  LINES  365 

8.  Reduce  the  equations  of  the  line 

2x  -Zy  -z  =  7,       5  ^-f  2z  -1  =  X 
to  the  symmetrical  form,  and  detei-mine  its  direction  cosines. 

9.  Find  the  angle  between  the  planes 

2a:— 3?/  —  2  =  7,      ^y-\-2z  —  l=x. 

10.  Find  the  angle  between  the  line 

X  +  y  +  2  z  =  0,     2x-y-2z-l=0, 
and  the  plane  2tx-\-Qz  —  by  +  l  =  0. 

11.  Write  the  equation  of  a  plane  parallel  to  the  plane 

2x-?/  +  7s-5  =  0, 
and  passing  through  the  point  (0,  0,  0) ;  through  the  point  (-1,  1,  -1). 

12.  Write  the  equation  of  a  plane  perpendicular  to  the  plane 

and  passing  through  the  two  points  (3,  1,  2)  and  (0,  ~2,  -4). 

13.  Find  the  distances  of  the  points  (7,  ~2,  3)  and  (3,  3,  1)  from  the 
plane  2a:  +  5?/  —  s  —  9  =  0.     Are  they  on  the  same  side  of  the  plane  ? 

14.  At  what  angle  does  the  plane  ax  +  hy  +  cz  -\-  d  =  0  cut  each  coordi- 
nate plane?     Each  coordinate  axis? 

15.  Find  the  equation  of  a  plane  through  the  point  (1,  1,  1)  and 
perpendicular  to  each  of  the  planes 

2a:-3?/  +  73  =  l,      x  -  y  -2z  =  2. 

16.  Write  the  equation  of  a  plane  whose  distance  from  the  point 
(0,  2,  1)  is  3,  and  which  is  perpendicular  to  the  radius  vector  of  the 
point  (2,  -1,  -1). 

17.  ^Vrite  the  equation  of  a  straight  line  through  the  point  (5,  2,  6) 
which  is  parallel  to  the  line 

2x-32  +  ?/-2  =  0,     a;  +  ?/  +  2  +  l=0. 

18.  Find  the  traces  on  the  coordinate  planes  of  tlie  line 

2a;-32  +  ?/-2  =  0,     x  +  y  +  z  +  l=Q. 

19.  Prove  that  the  planes 

2  a;  -  3  y  +  ^  +  1  =  0, 
5  a;  +  2  -  1  =  0, 
19a;-3?/-43-5  =  0, 
have  one  line  in  common. 


366  ANALYTIC  GEOMETRY  [Ch.  III.  224. 

20.  What  is  the  equation  of  the  plane  determined  by  the  line 

and  the  point  (5,  2,  6)  ? 

21.  Show  analytically  that  the  locus  of  a  point  equidistant  from  three 
given  points  is  a  straight  line  perpendicular  to  the  plane  determined  by 
those  three  points. 

22.  Derive  equation  [17]  directly  from  a  figure,  without  using  equa- 
tion [16  j. 


CHAPTER   IV 

EQUATIONS  OF  THE  SECOND  DEGREE 

QUADRIC   SURFACES 

225.  The  locus  of  an  equation  of  second  degree.  The  most 
general  algebraic  equation  of  second  degree  in  three  variables 
may  be  written 

Aoc^  +  By^  +  Cz^  +  2  Fyz  +  2  Gxz  +  2  Hocy  +  2  Xa?  +  2  3Iy 

+  2Nz  +  K  =  (i,  .  .  .  [31] 

Any  surface  which  is  the  locus  of  an  equation  of  second 
degree  is  called  a  quadric  surface,  and  is  of  particular 
interest  because  of  its  close  connection  with  and  analogy  to 
the  conic  sections.  In  fact,  every  plane  section  of  a  quadric 
is  a  conic,  as  may  be  easily  shown  as  follows. 

By  Art.  207,  any  plane  may  be  chosen  as  a  coordinate  plane, 
and  the  transformation  of  coordinates  to  the  new  axes  will 
leave  the  degree  of  equation  [31]  unchanged ;  i.e.^  the  new 
equation  of  the  locus  will  still  be  of  the  form  [31],  though 
with  different  values  for  the  coefficients.  To  find  the  nature 
of  any  plane  section,  choose  the  given  plane  as  (say)  the  xy- 
plane  of  reference,  and  transform  to  the  new  axes ;  the  new 
equation  will  be  of  form  (1).  Then  let  z  =  0.  The  equa- 
tion of  the  section  of  the  quadric  is 

Aa^  +  By^-^  'I Exy  +  2 Lx  +  2My  +  K=0;  .  .  (1) 

and  this,  by  Art.  175,  represents  a  conic. 

367 


368  ANALYTIC   GEOMETRY  [Ch.  IV. 

Moreover,  the  trace  of  the  surface  on  any  parallel  plane, 
as  2  =  «,  is  given  by  the  equation 

Ax^  +  Bi/  +  ^  Hxy  +  2(X  +  ah)x  +  2QM+  aF)y 

+  (CVH-2il[fa  +  ^)=0.       ...       (2) 

Now,  by  Arts.  177,  181,  the  loci  of  equations  (1)  and 
(2)  are  conies  of  the  same  species,  and  with  semi-axes  pro- 
portional; therefore  their  eccentricities  are  equal,  and  the 
curves  are  similar.  Hence,  all  parallel  plane  sections  of  a 
quadric  are  similar  conies. 

226.   Species  of   quadrics.     Simplified   equation   of   second 

degree.  As  will  be  seen  in  the  following  sections,  quadric 
surfaces  may  be  conveniently  classed  under  four  species. 
For,  although  different  plane  sections  of  any  surface  will  in 
general  be  conies  of  different  species,  still  the  general  form 
of  the  surface  may  be  characterized  most  strikingl}^  by  those 
plane  sections  which  are  ellipses,  hyperbolas,  parabolas,  or 
straight  lines.  These  species  are  called,  respectively,  ellip- 
soids^ hyperloloids,  paraboloids^  and  cones;  and  each  species 
has  special  varieties,  depending  upon  the  nature  of  a  second 
system  of  plane  sections.  To  study  these  species  it  will  be 
well  to  simplify  the  general  equation  of  second  degree  as 
much  as  possible  by  a  suitable  transformation  of  coordinates.* 
A  transformation  of  coordinates  changing  to  a  new 
rectangular  system  having  the  same  origin  as  the  old,  by 
equations  [14],  will  transform  the  given  equation  of  second 
degree  to 

A'x^  +  B^y'^  +  C'z^  +  2  F'yz  +  2  a^xz  +  2  Wxy  +  2  L^x 

-^2M'y+2N'z  +  K=0,       .       .       .       (1) 

where  A' ^  B\  "-  N'  are  functions  of  the  nine  direction  angles 


*  Compare  with  Art.  176. 


225-226.]  QUADRIC  SURFACES  369 

ctj,  «2i  •••  of  the  new  axes,  which  are  limited  by  the  six  inde- 
pendent equations  noted  in  Art.  207.  These  angles,  therefore, 
may  be  so  chosen  that  three  additional  conditions  shall  be 
fulfilled ;  hence,  so  that  the  coefficients  F',  G',  and  R'  shall 
vanish.     Then  the  new  equation  of  the  quadric  will  be 

A'x^-{-  B'i/-\-  Q'z^+  2  L'x  -f-  2  M'y  +  2  N'z  +K=  0.    (2) 

Now  a  second  transformation  may  be  made  to  a  parallel 
system  of  axes  through  a  new  origin  (A,  k^  y),  by  equations 
[13],  giving  for  the  new  equation 

A'x^  +  B'y'^  +  C'z^  +  2  L^'x  +  2  M"y  +  2N"z  +  K'  ^0,  (3) 

in  which  X'',  M" ^  N",  and  K'  are  functions  of  the  coordi- 
nates A,  ^,  and  j  ;  and  these  coordinates  may  be  chosen  so  that 
X",  M'\  and  N"  will  vanish,  giving  for  the  simplified  form 
of  the  equation  of  the  given  quadric, 

A'x^  +  B^y'^  +  C^z^  +  K'  =  ^.      .      .     .     (4) 

It  may  happen,  however,  that  the  choice  given  above  for 
the  direction  angles  ce^,  a^^  •••,  of  the  new  axes  is  such  that 
the  coefficient  of  one  more  term  of  second  degree,  as  (7',  will 
also  vanish  ;  then  equation  (4)  would  reduce  to 

A'x^  +  B'y'^  +  K'  =  0,      ...       (5) 

and  the  surface  is  a  cylinder  (Art.  210).  Again,  if  also  X", 
M" ^  N"  are  not  independent,  and  the  values  of  A,  ^,  j  as 
given  above  are  therefore  indeterminate,  then  7i,  ^,  j  may 
be  chosen  so  that,  for  example,  L",  M"^  and  K'  shall  vanish  ; 
and  the  equation  of  the  quadric  becomes 

Ax^^-B'y'^-^'lN"Z=0^      ...       (6) 

*  If  the  coefficients  of  two  quadratic  terms  vanish,  as  B'  and  6*',  a  change 
of  origin  first,  then  of  direction  of  axes,  may  be  chosen  so  that  the  equation 
will  reduce  to  the  form  (6). 

TAN.  AN.  GEOM. 24 


^2  = 


52=  _:^,  ^=  _^.     .     .     .     (1) 


370  ANALYTIC  GEOMETRY  [Ch.  IV. 

The  two  forms  of  tlie  quadric,  not  already  discussed,* 
have  therefore  for  their  equations,  when  simplified  (dropping 

the  accents), 

Ax'  +  Bf  +  Oz^  +  K=0,     .     .     .     [32] 

and  Ax^-^Bf  +  2M  =  0.       .       .       .       [33] 

A  center  of  a  surface  is  a  point  such  that  it  bisects  every 
chord  of  the  surface  which  passes  through  it.  It  is  clear 
that  the  locus  of  equation  [32]  is  a  central  quadric,  while 
the  locus  of  equation  [33]  is  non-central  (cf.  Art.  178). 

227.   Standard  forms  of   the  equation  of   a  quadric.     For 

convenience  of  discussion,  the  intercepts  of  the  locus  of 
equation  [32]  on  the  coordinate  axes  may  be  represented  by 
a,  b,  c,  respectively,  so  that 

T     "-    B'     '^-    o 

Then,  since  A,  B,  C,  and  K  cannot  be  all  of  the  same  sign, 
there  will  be  three  types  of  equation  [32],  according  to  the 
signs  of  A,  B,  (7,  and  K;   viz.: 

^2+52-^2-^'      •      •      •     y^^j 

a^      b^      c^~  *  •  •        ^^ 

Similarly,  equation  [33]  may  be  written  for  convenience  in 
the  typical  forms 

^2  ^^  j2  —  ^'  •        •        •       y'^J 

a2      62-^'  •  •         •        y'') 

*  An  exceptional  case  occurs  where  the  general  equation  can  be  factored 
into  linear  factors,  and  therefore  represents  two  planes. 


226-228.] 


QUADBIC  SURFACES 


371 


wherein,  however,  a  and  b  are  no  longer  intercepts  as  in 
(2),  (3),  and  (4). 

Again,  if  the  equation  [32]  has  its  constant  term  zero,  it 
may  be  written  in  two  typical  forms, 


a^     h^      c^ 


(8) 


These  seven  equations  are  standard  forms  of  the  equation 
of  second  degree,  and  will  be  discussed  in  turn. 

o  o  2 

228.  The  ellipsoid:  equation  ^  +  f-2  +  ^  =  l-     ^'^^^^  ^^® 
equation 

...       [34] 


^^^4.^  =  1 

a2  "^  &2  -^  c2 


the  following  properties  of  its  locus  may  be  derived  : 

(1)  The  traces  on  each  coordinate  plane  are  ellipses,  having 


Fig.  156 


372 


ANALYTIC  GEOMETRY 


[Ch.  IV. 


the  semi-axes  a  and  h  in  the  ir?/-plane,  h  and  c  in  the  yz- 
plane,  and  c  and  a  in  the  ;sa;-plane. 

(2)  The  traces  on  planes  parallel  to  any  coordinate  plane 
are  similar  ellipses  (Art.  225). 

(3)  The  equation  may  be  written 


y 


^2  ~^ 


=  1 


hence  for  a  plane  section  parallel  to  the  «/;3-plane,  the  semi- 
axes  are  real  if  the  value  of  x  lies  between  —  a  and  -{-  a, 
imaginary  if  beyond  those  limits,  and  zero  ii  x=  ±  a.  More- 
over, the  length  of  the  axes  diminish  continuously  from  the 
values  h  and  <?,  respectively,  when  a;  =  0  to  the  value  zero, 
when  X  =  ±  a. 

Similarly  for  sections  parallel  to  either  of  the  other 
coordinate  planes. 

(4)  The  surface  is  symmetrical  with  respect  to  each  co- 
ordinate plane. 

This  quadric  surface,  the  locus  of  equation  [34],  is  called 
an  ellipsoid.  It  may  be  conceived  as  generated  by  a  variable 
ellipse,  which  has  its  vertices  upon,  and  moves  always  per- 
pendicular to,  two  fixed  ellipses,  which  in  turn  are  perpen- 
dicular to  each  other  and  have  one  axis  in  common. 

From  this  definition  equation  [34]  can  be  easily  derived.  Let 
CRA  and  A  SB  be  fixed  ellipses  perpendicular  to  each  other,  and  having 

the  semi-axis  OA  in  common, 
and  the  second  axes  OC  and 
OB,  respectively;  and  let  SPR 
be  the  variable  ellipse,  with 
semi-axes  MS  and  MR.  If 
OA,  OB,  OC  he  taken  as  the 
X,  y,  z  axes,  respectively ;  and 
P  be  any  point  on  the  moving 
ellipse,  with  coordinates  OM, 
MM',  M'P,  then  (by  Art.  112), 


i 

z 

/ 

c 

— —^^^ 

/ 

0 

i 

1 

y 

1/ 

7 

s 

F 

IG.157 

u 


228-229.]  QUABRIC  SURFACES  373 


MR'       MS'        '  OC'      OA'        '  OB'       Ol'        ' 

MR'     MS'       ^  c^        a^       '     ^  ^  b^       a^  '   ■' 

By  equations  (2)  and  (3). 

Substitution  in  (1)  gives     E!  +  IC  -i.  ?^  =  1. 
Every  algebraic  equation  of  the  form 

represents  an  ellipsoid.  If  two  of  the  coefficients  of  the 
variable  terms  are  equal  it  is  an  ellipsoid  of  revolution, 
either  an  oblate  or  prolate  spheroid;  and  if  the  three  co- 
efficients of  the  variable  terms  are  equal,  it  is  a  sphere 
(cf.  Art.  213,  eqs.  (10),  (11),  and  (8)). 


229.   The  un-parted  hyperboloid  :  equation  -3+^ — 2^^' 


From  the  equation 


g  +  S-S=l      •        .        .         [85] 


the  following  properties  of  its  locus  may  be  derived  : 

(1)  The  trace  on  the  xt/  plane  is  an  ellipse,  with  semi-axes 
a  and  b  ;  while  the  traces  on  the  ^z-  and  zx  planes  are  hyper- 
bolas, having  the  semi-axes  b  and  c,  c  and  a,  respectively, 
and  the  conjugate  axes  along  the  2-axis. 

(2)  The  traces  on  planes  parallel  to  any  coordinate  plane 
are  similar  conies,  ellipses  or  hyperbolas,  respectively 
(Art.  225). 


374 


ANALYTIC  GEOMETRY 


[Ch.  IV. 


(3)    The  traces  on  the  planes  x  =  a^x=  —  a^  y  =  h^  y  =  —  h 
are  in  each  case  a  pair  of  intersecting  straight  lines. 


Fig.  158 


(4)    The  equation  may  be  written 


x^ 


+ 


r 


=1, 


•  (1) 


or 


y 


62(^2  _  ^2  j         C^{ofi  —  X^^ 


=  1. 


a' 


a^ 


.     (2) 


From  equation  (1)  it  appears  that  the  trace  on  the 
2;^ -plane  is  the  smallest  of  the  system  of  ellipses  parallel 
to  that  plane,  and  that  the  sections  increase  continuously 
and  indefinitely  as  z  increases  from  0  to  ±  oo. 
•  From  equation  (2)  it  appears  that  the  transverse  axis 
of  the  h}^erbolas  parallel  to  the  ^s-plane  is  parallel  to  the 
2/-axis.  Similarly  for  the  a:2-sections  the  transverse  axis 
is  parallel  to  the  a;- axis. 


229-230.] 


QUADRIC  SURFACES 


375 


(5)  The  surface  is  symmetrical  with  respect  to  each  co- 
ordinate plane.  '^" 

This  quadric  surface,  whose  equation  is  [35],  is  called  an 
un-parted  hyperboloid,  or  an  hyper boloid  of  one  sheet.  It 
may  be  conceived  as  generated  by  a  variable  ellipse,  which 
has  its  vertices  upon  and  moves  always  perpendicular  to  two 
fixed  hyperbolas,  which  in  turn  are  perpendicular  to  each 
other,  and  have  a  common  conjugate  axis.  Its  equation 
can  be  readily  obtained  from  this  definition.* 

Every  equation  of  the  form  Ax^  +  Bi/^  —  Cz^  —  IC=  0 
represents  an  un-parted  hyperboloid.  If  the  two  positive 
coefficients  are  equal,  i.e.^ii  a  =  h^  the  quadric  is  the  simple 
hyperboloid  of  revolution  (Art.  213,  eq.  (12)). 


230.   The  bi-parted  hyperboloid:    equation 

From  the  equation 

ar»2        y2        ^^  =  1 

a^     &2     ^2 


01,2       y2       Z^  _^ 
2       1,2      c'2         * 


a 


[36] 


the  following  properties  of  its  locus  may  be  derived : 


-X 


Fig.  159 


*Cf.  Art.  227. 


376  AJS'ALYTIC  GEOMETRY  [Ch.  IV. 

(1)  The  traces  on  the  xy-  and  2;a:-planes  are  hyperbolas, 
with  semi-axes  a  and  h,  c  and  a,  respectively,  and  with  the 
transverse  axis  along  the  x-axis,  while  the  traces  on  the 
planes  parallel  to  the  ?/;3-plane  are  imaginary  if  x  lies 
between  a  and  —  a,  real  ellipses  if  x  is  beyond  those  limits, 
and  points  \i  x  =  ±  a. 

(2)  The  traces  on  planes  parallel  to  any  coordinate  plane 
are  similar  (Art.  225). 

(3)  The  elliptical  sections  parallel  to  the  j/2!-plane  increase 
continuously  and  indefinitely  as  x  varies  from  +  «  to  +  oo, 
or  from  —  a  to  —  go. 

(4)  The  surface  is  symmetrical  with  respect  to  each 
coordinate  plane. 

This  quadric  surface,  whose  equation  is  [36],  is  called  a 
bi-parted  hyperboloid,  or  hyperboloid  of  two  sheets.  It  may 
be  conceived  as  generated  by  a  variable  ellipse  which  has 
its  vertices  upon,  and  moves  always  perpendicular  to,  two 
fixed  hyperbolas  which  in  turn  are  perpendicular  to  each 
other,  and  have  a  common  transverse  axis.  This  definition 
leads  readily  to  the  equation  [36]. 

Every  equation  of  the  form  Ax'^  —  By'^  —  Cz^  —  K—^  rep- 
resents a  bi-parted  hyperboloid.  If  the  coefficients  of  the 
two  negative  variable  terms  are  equal,  z.g.,  if  ^  =  c,  the  sur- 
face is  the  double  hyperboloid  of  revolution  (cf.  Art.  213, 
eq.  (13)). 

2  2 

231.  The  paraboloids:  equation  ^^^  =  ^'  A  discussion 
of  the  equation  ^^%  =  ^         *         *         *         t^'^] 

similar  to  that  of  the  preceding  articles  shows  that  its  locus 


230-231.] 


QUADRIC  SURFACES 


377 


is  as  represented  in  Fig.  160, 
symmetrical  with  respect  to 
the  yz-  and  ^a^-plane,  but  not 
with  respect  to  the  a^?/-plane. 
This  quadric  is  the  elliptic 
paraboloid,  and  may  be  con- 
ceived as  being  generated  by 
a  variable  parabola  which  has 
its  vertex  upon,  and  moves 
always  perpendicular  to,  a 
fixed  parabola,  the  axes  of  the  two  parabolas  being  parallel 
and  lying  in  the  same  direction.  This  definition  leads 
directly  to  equation  [37].* 

Every  equation  of  the  form  Ax^  -\-  B^^  —  2  Nz  =  0  repre- 
sents an  elliptic  paraboloid.  If  the  two  positive  coefficients 
are  equal,  the  quadric  is  a  paraboloid  of  revolution  (cf.  Art. 
213,  eq.  (14)). 

Similarly,  the  equation    ~^~jr~^       •  •         •         [^8] 


Fig.  160 


Fig. 161 


*  See  Art.  228. 


378 


ANALYTIC  GEOMETRY 


[Ch.  IV. 


has  for  its  locus  a  surface  as  represented  in  Fig.  161.  This 
quadric  is  the  hyperbolic  paraboloid,  and  may  be  conceived  as 
generated  by  a  variable  parabola  which  has  its  vertex  upon 
and  moves  always  perpendicular  to  a  fixed  parabola,  the  axes 
of  the  two  parabolas  being  parallel,  but  lying  in  opposite 
directions.  Equation  [38]  may  be  derived  at  once  from 
this  definition.* 

Every  equation  of  the  form  Ax^  —  By^  —  2  iV^  =  0  repre- 
sents an  hyperbolic  paraboloid. 


232. 


^i   ,   Vl      ^2 


The   cone:    equation    ^+57"^"^'     The   equation 


^     y^ 


+  —  =  0  evidently  is  sat- 


isfied  by  the  coordinates  of  only 
one  real  point,  viz.  the  origin. 
No  further  discussion  of  this 
equation  is  necessary.  But  the 
equation 


aj2      ^2      2;' 


-^=0 
c^ 


a'      l^     e^ t89] 

has  a  locus  of  importance,  hav- 
ing the  following  properties  : 

(1)  The  origin  is  a  point  of 
the  locus. 

(2)  The    trace    on    the    xy- 
plane  is  a   point.     The   traces 

on  planes  parallel  to  the  a7?/-plane  are  similar  ellipses,  whose 
semi-axes  increase  continuously  and  indefinitely  as  z  increases 
from  0  to  ±  CO. 

(3)  The  trace  on  each  of  the  other  coordinate  planes  is  a 
pair  of  straight  lines  which  intersect  at  the  origin. 


*  See  Art.  228. 


231-233.] 


QUADRIC  SURFACES 


379 


(4)  The  surface  is  symmetrical  with  respect  to  each  coordi- 
nate plane,  hence  also  with  respect  to  the  origin. 

(5)  The  straight  line  through  the  origin  and  any  other 
point  of  the  locus  lies  wholly  in  the  locus. 

This  quadric  surface  is  called  a  cone,  and  the  origin  is  its 
vertex.  It  may  be  conceived  as  generated  by  a  straight  line 
which  moves  along  a  fixed  ellipse  as  directrix,  and  passes 
through  a  fixed  point  in  a  straight  line  which  is  perpen- 
dicular to  the  plane  of  the  ellipse  at  its  center. 

Every  equation  of  the  form  Ax^  +  Bi/^  —  Cz^  =  0   repre 
sents  a  cone.     If  the  two  positive  coefficients  are  equal,  it  is 
a  cone  of  revolution,  or  circular  cone  (cf.  Art.  213,  eq.  (9)). 

The  reasoning  of  Art.  225,  applied  to  the  special  equation 
of  the  form  [31]  which  represents  a  cone,  gives  an  analytic 
proof  of  the  fact  that  every  plane  section  of  a  cone  is  a 
second  degree  curve  (cf .  Art.  48  j  Appendix,  Note  D). 

233.   The    hyperboloid    and    its  ^ 

asymptotic  cone.    The  hyperboloid 


X' 


a' 


and  the  cone 


a? 


yl 

62 


62 


=  1 


C2 


22 


are  closely  related.  It  is  clear 
that,  since  the  equations  differ 
only  in  the  constant  terms,  the 
surfaces  can  have  no  finite  points 
in  common  ;  while  as  the  values 
of  y  and  z  are  increased  indefi- 
nitely, the  corresponding  values 
for  X  from  the  two  equations  be- 


Ficbies 


380  ANALYTIC  GEOMETBT  [Ch.  IV.  233, 

come  relatively  nearer.  In  fact,  the  hyperboloid  may  be 
said  to  be  tangent  to  the  cone  at  infinity,  and  bears  to 
the  cone  a  relation  entirely  analogous  to  that  between 
the   hyperbola   and   its    asymptotes.       In    the    same    way, 

/T/.2  qj^  ly^ 

the  cone  -^  +  1^  — 7,=  ^  is  asymptotic    to   the   hyperboloid 
aP-      ¥      (T 

^_i_^_f!  =  1 

aP'      IP'      c^ 

EXAMPLES    ON    CHAPTER    IV 

1.  Derive  the  equation  [35]  directly  from  the  definition  of  Art.  229. 

2.  Derive  the  equation  [36]  directly  from  the  definition  of  Art.  230. 

3.  Derive  the  equations  [37],  [38]  directly  from  the  definitions  ol 
Art.  231. 

4.  Derive  the  equation  [39]  directly  from  the  definition  of  Art.  232 

5.  Show  analytically  that  the  intersection  of  two  spheres  is  a  circle. 

6.  Find  the  equation  of  the  tangent  plane  to  the  sphere  {x  —a)'^ 
-\-  (y  —  by^  -i-  (z  —  c)'^  =  r'^,  at  any  point  of  the  sphere. 

7.  Show  that  the  equation  Ax-^x  +  By^y  +  Cz^z  -{-  K  =  0  represents 
a  plane  tangent  to  the  conic,  Ax^  +  By'^  +  Cz^  ■}-  K  =  0,  at  the  point 
(Xj,  y^,  2j)  on  the  quadric. 

8.  Find  the  equation  of  the  cone  with  origin  as  vertex  and  the  ellipse 

h  —  =  1  in  the  plane  2  =  —  2,  as  directrix. 

9       4  ^ 

9.  Find  the  equation  of  a  sphere  having  the  line  from  Pi=  (a:^,  y^,  z^ 
to  f*2—  (-^^2'  2/2»  ^2)  ^^  ^  diameter. 

10.  Show  that  a  sphere  is  determined  by  four  points  in  space. 

Write  the  equation  of  the  quadric  whose  directing  curves  have  the 
equations : 

11.  ^'  +  ^'  =  1,    and    t^t^X. 
2       3  3       9 

12.  ^-^  =  1,    and    1  - -5- =  1. 
9       4  9       16 

13.  £'-^'  =  1,    and    ^'-^  =  1. 
94'  4      16 

14.  s2  _  16  X,    and    ?/2  =  9  x. 

15.  a;2  -  4  y  =  0,    and    ^2  _^  3  ?/  =  0. 


APPENDIX 

NOTE  A 

Historical  sketch.*  Analytic  Geometry,  in  the  form  in  which  it  is 
now  known,  was  invented  by  Rene  Descartes  (1596-1650)  and  first  pub- 
lished by  him  in  1637,  in  the  third  section  of  a  treatise  on  universal 
science  entitled  "  Discours  de  la  method  pour  bien  conduire  sa  raison  et 
chercher  la  verite  dans  la  sciences."  He  made  the  invention  while 
attempting  to  solve  a  certain  problem,  proposed  by  Pappus,  the  most 
important  case  of  which  is:  to  find  the  locus  of  a  point  such  that  the 
product  of  the  perpendiculars  drawn  from  it  upon  m  given  straight  lines 
shall  bear  a  constant  ratio  to  the  product  of  the  perpendiculars  drawn 
from  it  upon  n  other  given  straight  lines.  By  pure  geometry  this  prob- 
lem had  already  been  solved  for  the  special  cases  when  m  =  l  and  n  =  1 
or  2.  Pappus  had  also  asserted,  but  without  proof,  that  when  m  =  ?i  =  2, 
then  the  locus  of  this  point  is  a  conic.  In  his  effort  to  prove  this  fact 
Descartes  introduced  his  system  of  coordinates  and  found  the  equation 
of  the  locus  to  be  of  the  second  degree,  thus  proving  that  it  is  a  conic. 

Analytic  geometry  does  not  consist  merely  (as  is  sometimes  loosely 
said)  in  the  application  of  algebra  to  geometry :  that  had  been  done  by 
Archimedes  and  many  others,  and  had  become  the  usual  method  of  pro- 
cedure in  the  works  of  mathematicians  of  the  sixteenth  century.  But  in 
all  these  earlier  applications  a  special  set  of  axes  were  required  for  each 
individual  curve.  The  great  advance  made  by  Descartes  was  that  he 
saw  that  a  point  could  be  completely  determined  if  its  distances,  say  x 
and  ?/,  from  two  fixed  lines,  drawn  at  right  angles  to  each  other,  in  .the 
plane,  were  given  :  and  that  though  an  equation  f(x,  y)=0  is  indetermi- 
nate and  can  be  satisfied  by  an  infinite  number  of  values  of  x  and  y,  yet 
these  values  of  x  and  y  determine  the  coordinates  of  a  number  of  points 
which  form  a  curve  of  which  the  equation  /(x,  y)  =  0  expresses  some 
geometric  property,  i.e.,  a  property  true  for  every  point  of  the  curve. 
Moreover,  he  saw  that  this  method  enables  one  to  refer  all  the  curves 
that  may  be  under  investigation  to  the  same  set  of  axes ;   and  that  in 


*  Taken  chiefly  from  Ball's  History  of  Mathematics. 

381 


382 


APPENDIX 


order  to  investigate  the  properties  of  a  curve  it  is  sufficient  to  select  any- 
characteristic  geometric  property,  as  a  definition,  and  to  express  it  as  an 
equation  by  means  of  the  (current)  coordinates  of  any  point  on  the 
curve ;  i.e.,  to  translate  the  definition  into  the  language  of  analytic 
geometry  —  the  equation  so  obtained  contains  implicitly  every  property 
of  the  curve,  and  any  particular  property  can  be  deduced  from  it  by 
ordinary  algebra. 

While  the  earlier  geometry  is  an  admirable  instrument  for  intellectual 
training,  and  while  it  frequently  affords  an  elegant  demonstration  of 
some  proposition  the  truth  of  which  is  already  known,  it  requires  a 
special  procedure  for  each  individual  problem;  on  the  other  hand, 
analytic  geometry  lays  down  a  few  simple  rules  by  which  any  property 
can  be  at  once  proved.  It  is  incomparably  more  potent  than  the 
geometry  of  the  ancients  for  all  purposes  of  research. 


NOTE  B 

Construction  of  any  conic,  given  directrix,  focus,  and  eccentricity.  Let 
D'D  be  the  directrix,  F  the  focus,  and  e  the  eccentricity  of  a  conic 
(cf.  Part  I,  Art.  48),  to  plot  the  curve. 


Construction:  Draw  ZFX  perpendicular  to  D'D,  and  ZW so  that, 
if  a  =  ZXZW,  tan  a  =  e.  l^ow  draw  FR  perpendicular  to  ZF,  cutting 
ZW?itR]  then  it  is  a  point  of  the  conic  ;  it  is  the  end  of  the  latus  rectum. 

Bisect  the  right  angles  at  F  by  FR^  and  FH2,  intersecting  Z  W  in  H^ 
and  2^2'  ^^^^  draw  H^A  and  H^A'  perpendicular  to  ZX\  then  A  and  A' 
are  points  on  the  curve;  they  are  the  vertices  of  the  conic. 


APPENDIX 


383 


Again,  from  any  point  G  between  H^  and  H^  on  Z  W,  draw  MG  per- 
pendicular to  ZX,  cutting  it  at  M;  and  from  F  as  a  center  with  AIG 
as  radius  describe  an  arc  cutting  MG  at  P.  Then  P  is  a  point  of  the 
curve. 


Proof:  for  the  point  i?, 
for  the  point  A, 
for  the  point  P, 


FR      . 

—  =  tana  =  c; 

AF     AH,      . 

= ^  =  tan  a  =  e ; 

=  tan  a  =  e', 


ZA  ZA 
FP  ^  MG 
ZM     ZM 


[Z^Fi7i  =  45°] 


hence  the  points  R,  A^,  and  P  are  such  that  their  distances  from  the 
directrix  and  from  the  focus  are  in  the  ratio  e ;  and  each  is  therefore, 
according  to  the  definition  given  in  Art.  48,  a  point  of  the  conic.  By- 
plotting  various  points  P  (and  the  symmetrical  points  P')  and  connecting 
them  by  a  smooth  curve,  the  conic  may  be  plotted  to  any  requu^ed  degree 
of  accuracy. 

If  a<45°,  then  tana<l,  i.e.,  e  <  1,  and  the  conic  is  an  ellipse;  if 
a  =  45°,  the  conic  is  a  parabola ;  and  if  a  >  45°,  the  conic  is  an  hyperbola 
(cf.  Part  I,  Art.  48). 

NOTE  C 

The  special  cases  of  the  conies.  The  locus  of  the  second  degree  curve 
has  been  seen  to  have  three  species,  according  as  e<l,  e  =  1,  or  e>l. 

If  e  =  0,  then,  since  h  is  defined  by  the  equation  6"2  =  a^(l  —  e^),  b  =  a, 
and  the  curve  is  an  ellipse  with  equal  axes,  i.e.,  it  is  a  circle ;  in  this  case, 
also,  the  directrix  is  at  infinity  and  the  focus  at  the  center,  for  the  equa- 
tion of  the  directrix  is  x  =  -,  and  the  distance  from  the  center  to  the 

e 

focus  is  ae  (cf.  Part  I,  Arts.  110,  116). 


® 


384  APPENDIX 

Again,  suppose  the  focus  F  to  be  on  the  directrix.     Then,  if  P  is  any 
point  of  the  locus,  and  LP  perpendicular  to  FD, 

FP  =  e'LP,  .  .  .  (1) 

and  smZPFL=^  =  l;     ,  .  .  (2) 

l'  P      e 

hence  the  angle  PFL  is  constant,  with  two  supplementary  values  for  a 
given  value  of  e. 

The  locus  consists  therefore  of  two  straight  lines  intersecting  at  P, 
and  equation  (2)  shows  that : 

if  e  >  1,  the  lines  are  real  and  different ; 

if  e  =  1,  the  lines  are  real  and  coincident ; 

and    if  e  <  1,  the  lines  are  imaginary,  and  the  real  part   of  the  locus 
consists  of  the  point  F. 

Suppose  now  the  directrix,  with  the  focus  upon  it,  to  be  at  infinity ; 
then,  if  e  >  1,  the  locus  is  a  pair  of  parallel  lines. 

These  results  agree  with  those  akeady  summarized  in  Art.  182. 


NOTE  D 

Sections  of  a  cone  made  by  a  plane.  The  following  proposition  is 
due  to  Hamilton,  Quetelet,  and  others  (see  Taylor's  Ancient  and  Mod- 
ern Geometry  of  Conies,  p.  204). 

If  a  right  circular  cone  is  cut  by  a  plane,  and  two  spheres  are  inscribed 
in  the  cone  and  tangent  to  this  plane,  then  the  section  of  the  cone  made 
by  the  plane  is  a  second  degree  curve  (cf.  Part  T,  Arts.  48,  175),  of  which 
the  foci  are  the  points  of  contact  of  the  spheres  and  the  plane,  and  the 
directrices  are  the  lines  in  which  this  plane  intersects  the  planes  of  the 
circles  of  contact  of  the  spheres  and  the,  cone. 

Construction  :  Let  0-VW  be  a  right  circular  cone  cut  by  the  plane 
HK  in  the  section  RPSQ,  P  being  any  point  of  the  section.  Inscribe 
two  spheres,  C-ABF  and  C'-A'B'P,  whose  circles  of  contact  with  the 
cone  are  AEB  and  A'E'B',  respectively,  and  which  are  tangent  to  the 
plane  HK  in  the  points  F  and  F.  Through  P  draw  the  element  OP  of 
the  cone,  cutting  the  circles  of  contact  in  the  points  E  and  E'.  Also 
pass  a  plane  MN  through  the  circle  AEB,  and  therefore  perpendicular 
to  the  axis  OCC  oi  the  cone ;  it  will  intersect  the  plane  HK  in  a  straight 


APPENDIX 


385 


line  GDL,  which  is  perpendicular  to  the  straight  line  F'F.     Draw  PL 
perpendicular  to  GDL. 

P 


Then  PL  makes  a  constant  angle  0  (=Z  F'DA)  with  the  plane  MN 
[since  PL  is  parallel  to  F'F^,  and,  if  p  represents  the  distance  from  the 
point  P  to  the  plane  MN, 

p=PL  sin  e.  .  .  .  (1) 

Also  PE,  being  an  element  of  the  cone,  makes  a  constant  angle  a 
with  the  plane  MN,  and 

p  =PE  sin  a.  .  .  .  (2) 

Again,  since  tangents  from  an  external  point  to  a  sphere  are  equal, 

PE=PF.  .  .  .  (3) 

TAN.   AN.  GEOM. 25 


386  APPENDIX 

Hence,  from  equations  (1),  (2),  and  (3) 
PF     sin  " 


PL      sin  a 


=  e,  a,  constant,  .  ,  .  (4) 


i.e.,  the  ratio  PF :  PL,  for  every  point  P  of  the  section  SPRQ,  is  constant, 
and  (Part  T,  Arts.  48, 175)  the  section  is  a  second  degree  curve,  with  a  focus 

at  F,  directrix  GDL,  and  eccentricity  — . 

sm  a 

Similarly,  F'  is  the  other  focus,  and  the  line  of  intersection  of  the 
planes  HK  and  A'E'B'  is  the  other  directrix  of  the  conic  SPRQ;  hence 
the  theorem  is  established. 

Moreover,  the  plane  VW,  being  perpendicular  to  the  axis  of  the  cone, 
and  OVW,  being  a  section  made  by  a  plane  passing  through  the  axis, 
a  =  Z  OVW,  and  is  constant  for  a  given  cone,  while  0  =  Z  OSR,  and 
varies  only  with  the  plane  HK. 

Hence  the  eccentricity  varies  with  the  inclination  of  the  plane  HK, 
and  there  are  the  three  following  cases  : 

if  ^  <  a,  then  e  <  1,  and  the  section  is  an  ellipse ; 
if  ^  =  «,  then  e  =  1 ,  and  the  section  is  a  parabola ; 
if  ^  >  a,  then  e  >  1,  and  the  section  is  an  hyperbola. 

Again,  if  the  cutting  plane  HK  passes  through  the  vertex  0  of  the 
cone,  then  the  focus  F  is  on  the  directrix  GDL,  and  the  section  will  be 
either  a  pair  of  straight  lines  or  a  point : 

if  ^  <  a,  the  section  is  a  point,  the  vertex  0  of  the  cone. 

M  0  =  CL,  the  section  is  a  pair  of  coincident  straight  lines,  an  element  of 
the  cone ; 

ii6>  a,  the  section  is  a  pair  of  intersecting  straight  lines,  two  elements 
through  the  vertex  (cf.  Note  C). 

It  is,  of  course,  evident  that  for  every  elliptic  section  of  the  focal 
spheres  both  lie  in  the  same  nappe  of  the  cone,  and  touch  the  plane  of 
the  section  (HK)  on  opposite  sides;  while  for  every  hyperbolic  section 
these  focal  spheres  lie  one  in  each  nappe  of  the  cone,  and  both  on  the 
same  side  of  the  plane  of  the  section. 

In  the  above  proof,  for  the  sake  of  simplicity,  a  right  circular  cone 
was  employed;  it  is  easy  to  show  (see  Salmon's  Conic  Sections,  p.  329) 
that  every  section  of  a  second  degree  cone  (right  or  oblique)  by  a  plane 
is  a  second  degree  curve. 


APPENDIX 


387 


The  demonstration  just  given  shows  also  that  the  parabola  is  a  limit- 
ing case  of  an  ellipse  (cf.  Note  E).         ^_ 


NOTE  E 

Parabola  the  limit  of  an  ellipse,*  or  of  an  hyperbola.  If  a  vertex 
and  the  corresponding  focus  of  an  ellipse  remain  fixed  in  position  while 
the  center  moves  further  and  further  away,  the  major  axis  becoming 
infinitely  long,  then  the  form  of  the  ellipse  approaches  more  and  more 
nearly  to  that  of  a  parabola  having  the  same  vertex  and  focus. 


This  is  easily  shown  as  follows  : 

The  equation  of  the  ellipse  referred  to  its  major  axis  and  the  tangent 
at  its  left-hand  vertex,  as  coordinate  axes,  is  (Part  I,  Art.  112) 


{x  -  ay      if  _ 

«2  "^  62  ~  ■"' 


which  may  be  written  in  the  form 


2      2&2 
y^  = X 


62 


a  a^ 

If  now  the  fixed  distance  OF  be  represented  by  p,  then 
p  =  OF  =  OC  -  FC  =  a  -  \/«2  _  b% 
whence  h'^  =  2  ap  —  p'^ ; 

62       2  p      /)2 


(i) 


(2) 


therefore 


262  2»2 

=r  4  » — ,     and 

a  a  a^ 


a        a' 


*  This  fact  is  of  importance  in  astronomy  in  connection  with  the  behavior 
of  comets. 


388  APPENDIX 

Substituting  these  values  in  equation  (2)  it  becomes 

.^-(*^-'-i^>-(¥-a^'  •  •  •  (^) 

and  the  limit  of  this  equation  as  a  approaches  oo,  p  remaining  constant,  is 

?/2  =  4^a:;  .  .  .  (4) 

which  is  the  equation  of  a  parabola,  and  the  proposition  is  proved. 

In  the  same  way  it  may  be  shoAvn  that  the  parabola  is  the  limit  to 
which  an  hyperbola  approaches  when  its  center  moves  away  to  infinity, 
a  vertex  and  the  corresponding  focus  remaining  fixed  in  position 
(cf.  also  Note  D). 

NOTE   F 

Confocal  conies.  —  Two  conies  having  the  same  foci,  F^  and  Fg,  are 
called  confocal  conies.  Since  the  transverse  axis  of  a  conic  passes  through 
the  foci  and  its  conjugate  axis  is  perpendicular  to,  and  bisects,  the  line 
joining  the  foci,  therefore  confocal  conies  are  also  coaxial,*  i.e.,  they  have 
their  axes  in  the  same  lines.  If  the  equation  of  any  one  of  such  a  system 
of  conies  is 

and  if  X  is  an  arbitrary  parameter,  then  the  equation 

+  7^=1         ...  (2) 


a'^  +  X     h^  +  X 

will  represent  any  conic  of  the  system.  For,  a  and  b  being  constant,  and 
a'^b,  equation  (2)  represents  ellipses  for  all  values  of  X  between  co 
and  —  b%  hyperbolas  for  all  values  of  X  between  —  //^  and  —  a%  and 
imaginary  loci  when  X<  —  a^;  moreover,  the  distance  from  the  center 
0  to  either  focus  for  each  of  these  curves  is 

V(a2  +  A)  -  (62  +  X), 


which  equals  Va^  —  b^,  and  is  therefore  constant. 

The  individual  curves  of  the  system  represented  by  equation  (3)  are 
obtained  by  giving  particular  values  to  X,  each  value  of  X  determining 
one  and  but  one  conic.     If  any  one  of  these  conies  is  chosen  as  the 

*  Coaxial  conies  are,  however,  not  necessarily  confocal.  / 


APPENDIX 


389 


fundamental  conic,  and  represented  by  equation  (1),  then  each  of  the 
other  conies  of  the  system  may  be  designated  by  its  appropriate  vahie 
of  X. 


.^ 

^^ 

---^ 

/ 

^^N^/\      \ 

^ "" 

^*«*^/ 

/\y^ 

/xV'^ 

y\^ 

~~~^— =^Cl/H^ 

' y\ 

/    /Or* 

"^          '"" 

TttC 

'x  \ 

(  (\ 

\   "' 

1  ii 

\  \  Y^^--- 

Iw 

1^ 

"_Jj^ 

<y     ''^' 

-^ 

T,/ 

B> 

V 

Through  any  assigned  point,  P^  =  (a:j,  y^^  of  the  plane,  there  passes  one 
ellipse  and  one  hyperbola  of  the  system  represented  by  equation  (2). 
For  substituting  the  coordinates  x^  and  y^  of  P^  in  equation  (2),  it  gives 
the  quadratic  equation 


+ 


Vx 


+  A     &2  +  X 


=  1, 


(3) 


for  the  determination  of  X.  Equation  (3)  gives  two  values  of  A,  hence 
two  conies  of  this  confocal  system  pass  through  P^.  That  one  of  these 
is  an  ellipse  and  the  other  an  hyperbola  is  shown  as  follows :  the  quad- 
ratic function  in  A 


+ 


yi' 


a2  +  A     62  +  A 

is  negative  when  A  =  +  co,  and,  as  A  decreases  from  +  go  to  -co,  this 
function  becomes  positive  just  before  \--l\  negative  again  just  after 
\  =  -W-^  and  positive  again  just  before  A  =  -a'^;  hence,  of  the  two 
roots  of  equation  (3),  one  lies  between  -  }P-  and  co,  and  the  other  between 
—  a2  and  —  6^;  and  therefore  of  the  two  confocal  conies  which  pass 
through  Pj,  one  is  an  ellipse  and  the  other  an  hyperbola.  ^loreover,  the 
two  confocal  conies  which  pass  through  any  given  point,  as  P^  =  {x^,  ?/,), 
of  the  plane  intersect  at  right  angles.  This  is  easily  seen  geometrically 
thus:  connect  P^  with  the  foci  F^  and  P^,  then  the  tangent  P^T^  to  the 


390  AFFENBIX 

hyperbola  through  Py  bisects  the  interior  angle  between  F^P^  and  F^P^, 
while  the  tangent  P-^Tc,  to  the  ellipse  through  this  same  point  bisects  the 
external  angle  formed  by  these  two  lines  (cf.  Part  I,  Arts.  148,  163) ; 
these  tangents  are  therefore  at  right  angles,  hence  (cf.  Part  T,  Art.  100) 
the  conies  intersect  at  right  angles. 

This  fact  could  also  have  been  readily  proved  analytically  by  compar- 
ing the  equations  of  the  two  tangents. 

Remark  1.  It  is  easily  seen  that  as  X  approaches  —  h^  from  the  positive 
side,  the  ellipses  represented  by  equation  (2)  grow  more  and  more  flat 
(because  the  length  of  the  semi-minor  axis  VA-  +  A  approaches  0), 
approaching,  as  a  limit,  the  segment  F^F^  of  the  indefinite  straight  line 
through  the  foci.  On  the  other  hand,  if  X  approaches  —  h^  from  below, 
then  the  hyperbolas  grow  more  and  more  flat,  approaching,  as  a  limit, 
the  other  two  parts  of  this  line.  Again,  if  A,  approaches  —  a^  from 
above,  the  hyperbolas  approach  the  z^-axis  as  a  limit. 

Remark  2.  Since  through  every  point  of  a  plane  there  passes  one 
ellipse  and  one  hyperbola  of  the  confocal  system  represented  by  equation 
(2),  and  but  one  of  each,  therefore  the  two  values  of  X  which  determine 
these  two  curves  may  be  regarded  as  the  coordinates  of  this  point;  they 
are  known  as  the  elliptic  coordinates  of  the  point.  If  the  rectangular 
coordinates  of  a  point  are  known,  the  elliptic  coordinates  are  easily  found 
by  means  of  equation  (2). 

E.g.,  let  Pi=  {x^,  yi)  be  the  point  in  question,  then  the  elliptic  coor- 
dinates of  Pj  are  the  two  values  of  X,  which  are  the  roots  of  equation  (3). 
So,  too,  if  the  elliptic  coordinates  are  given,  the  Cartesian  coordinates  can 
be  found. 

Remark  3.  The  above  observations  concerning  confocal  conies  are 
easily  extended  to  confocal  quadrics,  i.e.,  to  quadric  surfaces  whose 
principal  sections  are  confocal  conies.  They  are  represented  by  the 
equation 

+  j!-  +  -^i-  =  i. 


+  A     b^-\-X     c2  +  A 


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